V ARIATION OF CURRENT WAVEFORM WITH TORQUE AND SPEED

The average electromagnetic torque is given by eqn. (4.14), and the energy-conversion loop area W is shown in Figs. 4.12 and 4.13. The objective of “average torque control” is a simple current pulse waveform which produces the required value of W corresponding to the torque demand. Even in simple cases, this is more complex than simply determining the required “value of current”, since the torque/ampere varies with both position and current. The following sections describe the general properties of the current waveform at different points in the torque/speed diagram, Fig. 4.35.

Low-speed motoring — At low speed the motor EMF e is low compared to the available supply voltage Vs, and the current can be regulated by chopping. If voltage-drops in the semiconductor devices are

neglected, the drive can apply three voltage levels +Vs, !Vs or 0 to the winding terminals to raise or

lower the flux and current. A simple strategy is to supply constant current throughout the torque zone, i.e., over the angle through which the phase inductance is substantially rising. Fig. 4.29 shows a typical low-speed motoring current waveform of this type in a 3-phase 6/4 motor at 500 rev/min.

The current i is chopped at about 8 A, starting 5E after the unaligned position (at 45E) and finishing 10E before the aligned position (at 90E). At first no torque is produced because the inductance is low and unchanging, but when the corners of the stator and rotor poles are within a few degrees of conjunction J, torque suddenly appears. It is controlled by the regulating the current. When the transistors are switched off, 10E before the aligned position, the current commutates into the diodes and falls to zero, reaching the “extinction” point a few degrees later, so that virtually no negative torque is produced. The flux-linkage R grows from zero and falls back to zero every stroke. When the driving transistors are first switched on, R grows linearly at first because the full supply voltage is applied across the winding terminals. When the current regulator starts to operate, R is also regulated to a constant value at first because the constant current is being forced into an inductance that is still almost constant at the low value around the unaligned position, before the poles begin to overlap. As soon as the pole corners approach conjunction J, the inductance starts to increase, so the flux-linkage R also increases as constant current is now being forced into a rising inductance. The flux-linkage continues to increase until the commutation point. After that, the diodes connect a negative “de-fluxing” voltage !Vs across the winding terminals and therefore R falls to zero very rapidly. In this example the

resistive voltage-drop is small, and therefore the rate of fall of flux-linkage is almost linear. At low speed the dwell is made approximately equal to $_s, since this is “width” of the “torque zone”, and this angle might typically be a little less than 30E in a typical 6/4 motor. De-fluxing is completed over only a small angle of rotation since the speed is low, so the entire conduction stroke occupies only about 30E.

Fig. 4.30 High-speed motoring waveforms.

The process is summarized in the energy-conversion loop, which fits neatly between the aligned and unaligned magnetization curves as a result of the selection of the firing angles. It appears that the energy conversion W could be increased slightly, by retarding the commutation angle to extend the loop up to the aligned magnetization curve. This would not require any increase in peak current, but it would increase the average and r.m.s. values. It is also possible that delayed commutation could incur a period of negative torque just after the aligned position, which would appear as a re-entrant distortion of the energy-conversion loop, limiting the available gain in torque. Operation is at point M1 in the torque/speed characteristic, Fig. 4.35. It is possible to maintain torque constant with essentially the same current waveform as the speed increases up to a much higher value, since the motor EMF is still much lower than the supply voltage.

High-speed motoring — At high speed the motor EMF is increased and the available voltage may be insufficient for chopping, so that the torque can be controlled only by varying the firing angles of a single pulse of current. Fig. 4.32 shows a typical example, in which the speed is 1300 rev/min.

The driving transistors are switched on at 50E and off at 80E, the same as in Fig. 4.29. At first the overlap between poles is small, and the supply voltage forces an almost linear rise of current di/dt =

Vs/Lu into the winding. Just before the start of overlap the inductance begins to increase and the back-

EMF suddenly appears, with a value that quickly exceeds the supply voltage and forces di/dt to become negative, making the current fall. The higher the speed, the faster the current falls in this region. Moreover, for a given motor there is nothing that can be done to increase it, other than increasing the supply voltage. The torque also falls. Operation is at point M2 in Fig. 4.35.

Operation at much higher speed — At a certain “base speed” the back-EMF rises to a level at which the transistors must be kept on throughout the stroke in order to sustain the rated current. Any chopping would reduce the average applied voltage and this would reduce the current and torque. The “base” speed is marked B in Fig. 4.35. If resistance is ignored, the peak flux-linkage achieved during the stroke is given by V_s)2/T, where )2 is the “dwell” or conduction angle of the transistors. If the peak flux is to be maintained at higher speeds, the “dwell” must be increased linearly with speed above the base speed. At high speed the turn-on angle can be advanced at least to the point where the sum of the fluxing and de-fluxing intervals is equal to the rotor pole-pitch, at which point conduction becomes continuous (i.e. the current never falls to zero). This corresponds to a dwell of 45E and a total conduction stroke of 90E, neglecting the effect of resistance (which tends to shorten the de-fluxing interval).

Fig. 4.31 Very high speed motoring

Fig. 4.32 Energy-conversion loops at low and high speed, 1300 and 3900 rev/min.

Thus it appears that the dwell or “flux- building angle” can increase from 30E at low speed to 45E at high speed, an increase of 50% or 1.5:1. Over a speed range of 3:1, the peak flux-linkage might therefore fall to 1.5/3 = 0.5, or one-half its low-speed value. This is illustrated in Fig. 4.31 for a speed of 3900 rev/min. The peak current is approximately unchanged but the loop area

W is only about one-third of its low-speed

value. The comparison between the loop areas at 1300 and 3900 rev/min is shown more clearly in Fig. 4.32. The average torque is therefore only about one-third of its low-speed value, but the power remains almost unchanged. Operation is at point M3 in Fig. 4.35.

Low-speed generating — Low-speed generating is similar to low-speed motoring except that the firing angles are retarded so that the current pulse coincides with a period of falling inductance. Fig. 4.33 shows a typical example. The average torque is negative and the energy-conversion loop is traversed in the clockwise direction. At the start of the stroke, there is a slight positive torque because the current is switched on shortly before the aligned position, while the inductance is still rising. In this example the torque falls to zero before the current is commutated, indicating that the commutation angle could be advanced slightly without reducing the average torque. The reduction in copper loss would increase the efficiency. During that “tail” period when there is current but no torque, the current is maintained by the drive which is simply exchanging reactive energy with the DC link filter capacitor. Operation is at G1 in Fig. 4.35.

Fig. 4.33 Low-speed generating waveforms

High-speed generating — High-speed generating is similar to high-speed motoring, except that the firing angles are retarded so that the current pulse coincides with a period of falling inductance. Fig. 4.34 shows a typical example. The torque is negative and the energy-conversion loop is again traversed in the clockwise direction. At the start of the stroke, there is a slight positive torque because the current is switched on a few degrees before the aligned position, while the inductance is still rising. Operation is at G2 in Fig. 4.35.

Operating regions — torque/speed characteristic

For control purposes the torque/speed envelope can be divided into regions as shown in Fig. 4.35.

Constant torque region—The base speed is the maximum speed at which maximum current and rated torque can be achieved at rated voltage. In this region the torque is controlled by regulating the current, with relatively minor adjustments in the firing angles as necessary to alleviate noise or improve the current or torque waveform, or to improve efficiency.

Fig. 4.35 Torque-speed characteristics

Constant power region—As the speed and back-EMF increase, the dwell is increased to maintain the peak flux-linkage at the highest possible level. If the dwell is equal to half the rotor pole-pitch and the de-fluxing angle is negligible at the base speed, then in principle the dwell can be doubled before the onset of continuous conduction. Therefore if the dwell is increased in proportion to speed, the peak flux- linkage can be maintained up to about twice the base speed. However, constant power can be maintained to a higher speed than this, because the loss of loop area dW/dT is compensated by the increase in speed. If power is taken as TT and T % W, then P % TW and for constant power we require that )P = T)W + W)T = 0, which says that constant power can be maintained up to the point where )W/W = !T/)T. In other words, the maximum speed at constant power is the speed at which the rate

of loss of loop area is balanced by the rate of increase of speed. The rate of increase in back-EMF is less than proportional to the speed, because the current decreases with speed and MR/M2 is reduced. (In the linear analysis e = iT dL/d2, and i is decreasing while T is increasing and dL/d2 remains constant.

Falling power region—Eventually as the speed increases, the turn-on angle can be advanced no more, and the torque falls off more rapidly so that constant power cannot be maintained, even though very high speeds can be attained against a light load. The maximum phase advance depends on the drive controller. If the turn-on angle is advanced beyond the point where the dwell becomes equal to about half the rotor pole-pitch, continuous conduction will begin: the phase current never falls to zero and the energy-conversion loop “floats” away from the origin. As it does so, it moves to a region where the separation between the aligned and unaligned curves is increased, and the torque per ampere actually increases. For this reason, operation with continuous conduction is a possible means of increasing the power density, not only at high speeds but even at low speeds. The increase in copper loss is acceptable if there is a greater gain in converted power and the machine can withstand the temperature rise. A similar effect can be achieved with a DC bias winding in 3-phase motors, [Horst, 1995].

Reversibility—Fig. 4.35 shows only two quadrants of the torque/speed characteristic, corresponding to motoring and generating (or braking). The direction of rotation is the same in both quadrants. Operation in the opposite direction is symmetrical, provided that the rotor position transducer can provide the correct reference position and direction sense. The firing angles for motoring in one direction become generating angles in the reverse direction, at least at low speed. The machine is thus reversible and regenerative, and able to operate in all four quadrants of the torque/speed diagram.

Multiple-phase operation — To produce torque at all rotor positions the entire 360E of rotation must be ‘covered’ by segments of rising inductance from different phases, as shown in Fig. 4.16, and the phase currents must be sequenced to coincide with the appropriate segments. The total torque averaged over one revolution is usually assumed to be the sum of the torque contributions from each phase. Although the calculation and control of torque are both referred to one phase, some degree of overlap is required in practice to minimise notches in the instantaneous torque waveform when the phases are commutated, and to produce adequate starting torque at all rotor positions.

Fig. 4.36 Conduction modes