1. ECE 8830 - Electric Drives Topic 13: Vector Control of AC Induction
Motors
Spring 2004

2. Introduction
Scalar control of ac drives produces good steady state performance but poor dynamic response. This manifests itself in the deviation of air gap flux linkages from their set values. This variation occurs in both magnitude and phase.
Vector control (or field oriented control) offers more precise control of ac motors compared to scalar control. They are therefore used in high performance drives where oscillations in air gap flux linkages are intolerable, e.g. robotic actuators, centrifuges, servos, etc.

3. Introduction (cont’d)
Why does vector control provide superior dynamic performance of ac motors compared to scalar control ?
In scalar control there is an inherent coupling effect because both torque and flux are functions of voltage or current and frequency. This results in sluggish response and is prone to instability because of 5th order harmonics. Vector control decouples these effects.

4. Torque Control of DC Motors
There is a close parallel between torque control of a dc motor and vector control of an ac motor. It is therefore useful to review torque control of a dc motor before studying vector control of an ac motor.

5. Torque Control of DC Motors (cont’d)
A dc motor has a stationary field structure (windings or permanent magnets) and a rotating armature winding supplied by a commutator and brushes. The basic structure and field flux and armature MMF are shown below:

6. Torque Control of DC Motors (cont’d)
The field flux f (f) produced by field current If is orthogonal to the armature flux a (a) produced by the armature current Ia. The developed torque Te can be written as:
Because the vectors are orthogonal, they are decoupled, i.e. the field current only controls the field flux and the armature current only controls the armature flux.

7. Torque Control of DC Motors (cont’d)
DC motor-like performance can be achieved with an induction motor if the motor control is considered in the synchronously rotating reference frame (de-qe) where the sinusoidal variables appear as dc quantities in steady state.
Two control inputs ids and iqs can be used for a vector controlled inverter as shown on the next slide.

8. Torque Control of DC Motors (cont’d)
With vector control:
ids (induction motor)  If (dc motor)
iqs (induction motor)  Ia (dc motor)
Thus torque is given by:
where is peak value of sinusoidal space vector.

9. Torque Control of DC Motors (cont’d)
This dc motor-like performance is only possible if iqs* only controls iqs and does not affect the flux , i.e. iqs and ids are orthogonal under all operating conditions of the vector-controlled drive.
Thus, vector control should ensure the correct orientation and equality of the command and actual currents.

10. Equivalent Circuit of Induction Motor
The complex de-qe equivalent circuit of an induction motor is shown in the below figure (neglecting rotor leakage inductance).

11. Equivalent Circuit of Induction Motor (cont’d)
Since the rotor leakage inductance has been neglected, the rotor flux = , the air gap flux.
The stator current vector Is is the sum of the ids and iqs vectors. Thus, the stator current magnitude, is related to ids and iqs by:

12. Phasor Diagrams for Induction Motor
The steady state phasor (or vector) diagrams for an induction motor in the de-qe (synchronously rotating) reference frame are shown below:

13. Phasor Diagrams for Induction Motor (cont’d)
The rotor flux vector is aligned with the de axis and the air gap voltage is aligned with the qe axis. The terminal voltage Vs slightly leads the air gap voltage because of the voltage drop across the stator impedance. iqs contributes real power across the air gap but ids only contributes reactive power across the air gap.

14. Phasor Diagrams for Induction Motor (cont’d)
The first figure shows an increase in the torque component of current iqs and the second figure shows an increase in the flux component of current, ids. Because of the orthogonal orientation of these components, the torque and flux can be controlled independently. However, it is necessary to maintain these vector orientations under all operating conditions.
How can we control the iqs and ids components of the stator current Is independently with the desired orientation ?

15. Principles of Vector Control
The basic conceptual implementation of vector control is illustrated in the below block diagram:
Note: The inverter is omitted from this diagram.

16. Principles of Vector Control (cont’d)
The motor phase currents, ia, ib and ic are converted to idss and iqss in the stationary reference frame. These are then converted to the synchronously rotating reference frame d-q currents, ids and iqs.
In the controller two inverse transforms are performed:
1) From the synchronous d-q to the
stationary d-q reference frame;
2) From d*-q* to a*, b*, c*.

17. Principles of Vector Control (cont’d)
There are two approaches to vector control:
1) Direct field oriented current control
- here the rotation angle of the iqse vector with respect to the stator flux qr’s is being directly determined (e.g. by measuring air gap flux)
2) Indirect field oriented current control
- here the rotor angle is being measured indirectly, such as by measuring slip speed.

18. Direct Vector Control
In direct vector control the field angle is calculated by using terminal voltages and current or Hall sensors or flux sense windings.
A block diagram of a direct vector control method using a PWM voltage-fed inverter is shown on the next slide.

19. Direct Vector Control (cont’d)

20. Direct Vector Control (cont’d)
The principal vector control parameters, ids* and iqs*, which are dc values in the synchronously rotating reference frame, are converted to the stationary reference frame (using the vector rotation (VR) block) by using the unit vector cose and sine. These stationary reference frame control parameters idss* and iqss* are then changed to the phase current command signals, ia*, ib*, and ic* which are fed to the PWM inverter.

21. Direct Vector Control (cont’d)
A flux control loop is used to precisely control the flux. Torque control is achieved through the current iqs* which is generated from the speed control loop (which includes a bipolar limiter that is not shown). The torque can be negative which will result in a negative phase orientation for iqs in the phasor diagram.
How do we maintain idsand iqs orthogonality? This is explained in the next slide.

22. Direct Vector Control (cont’d)

23. Direct Vector Control (cont’d)
Here the de-qe frame is rotating at synchronous speed e with respect to the stationary reference frame ds-qs, and at any point in time, the angular position of the de axis with respect to the ds axis is e (=et).
From this phasor diagram we can write:
and

24. Direct Vector Control (cont’d)
Thus,
, , and
The cose and sine signals in correct
phase position are shown below:

25. Direct Vector Control (cont’d)
These unit vector signals, when used in the vector rotation block, cause ids to maintain orientation along the de-axis and the iqs orientation along the qe-axis.

26. Summary of Salient Features of Vector Control
A few of the salient features of vector control are:
The frequency e of the drive is not controlled (as in scalar control). The motor is “self-controlled” by using the unit vector to help control the frequency and phase.
There is no concern about instability because limiting within the safe limit automatically limits operation to the stable region.

27. Summary of Salient Features of Vector Control (cont’d)
Transient response will be fast because torque control by iqs does not affect flux.
Vector control allows for speed control in all four quadrants (without additional control elements) since negative torque is directly taken care of in vector control.

28. Flux Vector Estimation
The air gap flux can be directly measured in a machine using specially fitted search coils or Hall effect sensors. However, the drift in the integrator with a search coil is problematic at very low frequencies. Hall effect sensors tend to be temperature-sensitive and fragile.
An alternative approach is to measure the terminal voltage and phase currents of the machine and use these to estimate the flux. These techniques are discussed on pp of the Bose text.

29. Indirect Vector Control
Indirect vector control is similar to direct vector control except the unit vector signals (cose and sine) are generated in a feedforward manner.
The phasor diagram on the next slide can be used to explain the basic concept of indirect vector control.

30. Indirect Vector Control (cont’d)

31. Indirect Vector Control (cont’d)
The ds-qs axes are fixed on the stator and the dr-qr axes are fixed on the rotor. The de-qe axes are rotating at synchronous speed and so there is a slip difference between the rotor speed and the synchronous speed given by:
Since, , we can write:

32. Indirect Vector Control (cont’d)
In order to ensure decoupling between the stator flux and the torque, the torque component of the current, iqs, should be aligned with the qe axis and the stator flux component of current, ids, should be aligned with the de axis.
We can use the de-axis and qe-axis equivalent circuits of the motor (shown on the next slide) to derive control expressions.

33. Indirect Vector Control (cont’d)

34. Indirect Vector Control (cont’d)
The rotor circuit equations may be written as:

35. Indirect Vector Control (cont’d)
The rotor flux linkage equations may be written as:
These equations may be rewritten as:

36. Indirect Vector Control (cont’d)
Combining these with the earlier equations allows us to eliminate the rotor currents which cannot be directly obtained. The resulting equations are:
where

37. Indirect Vector Control (cont’d)
For decoupling control the total rotor flux needs to be aligned with the de-axis and so we want: qr=0 => dqr/dt =0
If we now substitute into the previous equations, we get:
and
where has been substituted for dr .

38. Indirect Vector Control (cont’d)
For implementing the indirect vector control strategy, we need to take these equations into consideration as well as the equation:
Note:
A constant rotor flux results in the equation:
so that the rotor flux is directly proportional to ids in steady state.

39. Indirect Vector Control (cont’d)
An implementation of indirect vector control for 4-quadrant operation is shown below:

40. Indirect Vector Control (cont’d)
Features of this implementation:
Diode rectifier front-end with a PWM inverter with a dynamic brake in the dc link.
Hysteresis-band current control.
Speed control loop generates the torque component of current, iqs*.
Constant rotor flux is maintained by using the desired ids*.
The slip frequency sl* is generated from the desired iqs*.

41. Indirect Vector Control (cont’d)
Slip gain Ks is given by:
e and e are given by:
and
The incremental encoder is necessary for indirect vector control because the slip signal locates the rotor pole position with respect to the dr axis in a feedforward manner.

42. Indirect Vector Control (cont’d)
If iqs*<0 for negative torque, phasor iqs is reversed and sl (and sl) will be negative.
The speed control range can be extended into the field weakening region by incorporating the dotted line part of the implementation (see figure below). Note: Closed loop flux control is now required.

43. Indirect Vector Control (cont’d)
Harmonic content of hysteresis-band current control is not optimum. Also, at higher speeds the current controller will saturate in part of the cycle because of the high back emf.
Synchronous current control can be used to overcome these problems. See Bose text, pp for details.

44. Indirect Vector Control (cont’d)
A dc motor-like electromechanical model can be derived for an ideal vector-controlled drive using the following equations:

45. Indirect Vector Control (cont’d)
A transfer function block diagram is shown below:
Note: The torque Te responds instantly but the flux has first order delay (with time constant =Lr/Rr).

46. Indirect Vector Control (cont’d)
The physical principle of vector control can be explained more clearly with the help of the below de-qe equivalent circuits:

47. Indirect Vector Control (cont’d)
Since ids and iqs are being controlled, we can ideally ignore the stator-side parameters. With qr=0 under all conditions, the emf source on the rotor side de-circuit slqr=0. This means that in steady state ids flows only through the magnetizing inductance, Lm, but in the transient case, is shared by the rotor circuit whose time constant = Llr/Rr.

48. Indirect Vector Control (cont’d)
In the qe-circuit when torque is controlled by iqs the emf sldr changes instantaneously (because ). Since qr=0, this emf causes a current (Lm/Lr)iqs to flow through the rotor resistor Rr. If Llr is neglected and flux is constant, ids is seen to only flow through Lm and iqs only flows through the rotor side, as desired.

49. Indirect Vector Control (cont’d)
A serious issue with respect to indirect vector control is that of slip gain detuning. This is due primarily to variation in rotor resistance. This effect is illustrated below where Rr=actual rotor resistance and
= estimated rotor resistance.

50. Indirect Vector Control (cont’d)
Continuous on-line tuning of Ks is very complex and computationally intensive. However, two methods, one based on extended Kalman filtering (EKF) for parameter estimation and a second one based on a model referencing adaptive controller (MRAC) approach are good options. The EKF method will be considered later when studying sensorless vector control but the MRAC method is described next.

51. Indirect Vector Control (cont’d)
In the MRAC approach a reference model output signal X* that satisfies the tuned vector control condition is usually a function of ids* and iqs*, motor inductances, and operating frequency. The adaptive model X is estimated based on motor feedback voltages and currents as shown in the next slide. X is compared to X* an the resulting error used to estimate the slip gain through a P-I compensator. Slip gain tuning is achieved when X=X*.

52. Indirect Vector Control (cont’d)

53. Indirect Vector Control (cont’d)
Suppose we decide to use torque as the model parameter X. Thus,
Substituting Lmids* for gives:
The actual torque can be estimated from the stator frame variables using the equation:

54. Indirect Vector Control (cont’d)
Note: Lm and Lr parameter variations affect the estimation accuracy of X* and at low speeds, the stator resistance Rs affects the estimation accuracy of X.

55. Stator Flux-Oriented Vector Control
Until now we have only considered rotor flux-oriented vector control. Airgap flux or stator flux-oriented vector control is also possible but at a cost of a coupling effect that requires decoupling compensation. See Bose text pp for details.

56. Vector Control of Current-Fed Inverter Drive
Vector control can also be extended to current-fed drives as illustrated below:

57. Vector Control of Current-Fed Inverter Drive (cont’d)
Drive operates with regulated rotor flux and the speed control loop is the outer loop. The speed loop generates the torque command Te* which is then divided by K to generate iqs*. The flux loop generates ids*.
is used to control the firing angle of the phase controlled rectifier through a feedback loop. The inverter frequency is controlled by a phase-locked loop (PLL) so that the stator current, , is maintained at the desired torque angle with respect to the rotor flux.

58. Vector Control of Cycloconverter Drive
Vector control can also be used with a Scherbius drive with cycloconverter as shown:

59. Vector Control of Cycloconverter Drive (cont’d)
Recall in the Scherbius drive, +sPg is sent to the line in subsynchronous motoring and -sPg is sent to the line in supersynchronous motoring, where sPg is the slip energy. Currents Ip and IQ are the in-phase and quadrature current components, with respect to the slip voltage, Vr. The error from the speed control loop generates the desired current, Ip*’ and IQ* may be set to zero (as shown).

60. Vector Control of Cycloconverter Drive (cont’d)
The unit vector signals are obtained from the following equations:
where and ,
and is the amplitude of the line voltage.

61. Vector Control of Cycloconverter Drive (cont’d)
To illustrate how the drive works, consider a drive that is accelerating from a subsynchronous speed with a command supersynchronous speed.
At subsynchronous speed:
IP >0, sl >0, and sPg>0.
At synchronous speed, sl=0 and IP is dc.
At supersynchronous speed:
IP <0, => sl <0, and sPg<0.