Mechanism of Torque Production

34.8.3 Mechanism of Torque Production

34.8.3.1 Variable-reluctance Motor

If it is assumed that the current in the excited wind- where P is the number of rotor poles.

ing remains constant, the production of static torque of a variable-reluctance motor around a detent position is given by

34.8.2.4 Permanent-magnet Stepping Motor

dW m

Permanent-magnet stepper motors have alternate polarities of

dθ

permanent magnets on the rotor surface while the rotor iron, if it is used, has no teeth. In one type of construction, the

This torque expression may also be expressed as in Eq. 34.94, rotor has no iron, and the stator consists of two windings that when it is further assumed that the inductance of the

34 Motor Drives 969 winding may be reversed by the direction of its current. Con-

sequently, the polarity of the winding currents also determines the direction in which the developed torque increases positively around a detent position.

FIGURE 34.87 Stator and rotor teeth alignment: (a) aligned position,

34.8.4 Single- and Multi-step Responses

θ = X and (b) unaligned position, θ = Y . When the rotor is at a detent position and phase currents are

changed to a new value, the detent position is moved and excited winding at any given position remains constant for all the rotor proceeds towards it and settles down at the new currents.

detent position. The movement of the rotor is influenced by the shape of the T–θ characteristic and the load friction. The

rotor stepping is normally quite under-damped. The final posi- T = i

tioning error is also determined largely by the load torque. For instance, if the T–θ characteristic is assumed to be a sinusoidal

The developed torque is due to the variation of inductance function of θ, the error in stepping is given by Eq. (34.95), (or reluctance) with position. Note that the direction of cur- where T max is the peak of the T–θ characteristic and T L is the rent has no bearing on the developed torque. When the stator load friction torque. and rotor poles are perfectly aligned, as indicated in Fig. 34.87a, the inductance L changes little with a small change in θ. The

θ e = sin −1 ( T L /T max ) (34.95) developed torque is thus very small around this position, corre- sponding to the position X in Fig. 34.82. When the stator and

However, this error does not accumulate as further stepping rotor teeth are unaligned, as in Fig. 34.87b, L changes more is performed. If the phase currents are switched in succession, significantly with θ, and the restoring torque becomes much the rotor makes multiple steps. Typical single and multi step larger. As θ increases, dL/dθ goes through a maximum, pro- responses are as indicated in Fig. 34.88. ducing T max . It should be noted that around a stable detent,

The maximum rate at which the rotor can be moved L reduces as θ increases, so that the slope of the T–θ char- depends on several factors. The rise and fall times of the wind- acteristic is negative at the origin. Beyond the position where ing currents, which are largely determined by the electrical T max is developed, L increases as a result of the next set of parameters of the windings and the type of drive circuits used, rotor teeth coming under the stator teeth. This explains the and the combined inertia and friction parameters of the motor drop in T max and the positive slope of the T–θ characteris- and load are important factors. tic in the region between where T max is developed and Y in

The discrete signals to step the motor in the forward or Fig. 34.82.

reverse direction are translated into current-switching signals If stepper motors are operated in magnetically linear region for the drive circuits. This translator is a simple logical oper- where L remains constant with the current for a given angu- ation that is embedded in most of the integrated circuits lar position, the developed torque per unit volume is small. available for driving stepper motors. Because of this, steppers motors are normally driven far

In many applications, the stepper motor is operated at far into saturation. Equation (34.94) then does not represent the higher speeds than which it can start/stop from. The perfor- torque characteristic adequately.

mance of a stepper motor at high speed is normally given For a saturated stepper motor, the calculation of the T –θ in terms of its pull-out torque-speed (T–ω) characteristic. characteristic for any given current involves complex compu- tation of stored energy, or coenergy, for each position of the rotor. This requires the magnetization characteristics of the motor for different levels of stator currents and rotor posi- tions to be known. Reference [33] may be consulted for further reading on this.

34.8.3.2 Hybrid and PM Motors

In hybrid stepper motors, most of the developed torque is con- tributed by the variable-reluctance principle explained earlier. The rest is developed by the rotor magnet in striving to find

(b) the minimum-reluctance position. It should be noted that FIGURE 34.88 Typical step responses of a stepper motor: (a) single step

(a)

the alternate polarities of the magnetic poles created by each response and (b) multi step response.

970 M. F. Rahman et al. Pull-out

34.8.5.1 Unipolar Drive Circuits

torque, Nm

In its simplest form, the unipolar drive circuits, one for each winding, are as indicated in Fig. 34.90. The transistor (MOSFET) is turned on to energize the winding, with a current that is limited either by the winding resistance or by hysteresis or PWM current controllers. The freewheeling diode allows the winding current a circulating path when the transistor is

turned off.

The drive circuit of Fig. 34.90a is a basic one. A better drive FIGURE 34.89 Typical pull-out torque characteristic of a stepper

Stepping rate in steps/sec

circuit is shown in Fig. 34.90b, which includes a zener diode motor.

in the freewheeling path. A pulse-width modulator is also included in the gate driving circuit. The pulse-width modu- lator allows a higher dc supply voltage (typically 5–10 times the voltage for the resistance-limited drive) to be used, thereby

This characteristic indicates the maximum average torque, the reducing the rise time of current at switch-on by 5–10 times. motor may develop while stepping continuously at a given rate. The zener diode allows a fast fall time for the current when the This torque is also largely determined by the parameters of the transistor is turned off by dissipating the trapped energy of the motor and its drive circuits. Figure 34.89 indicates the typical winding at switch-off faster. Yet another scheme is shown in shape of the pull-out T–ω characteristic of a stepper motor Fig. 34.90c which allows the trapped energy of the winding at drive.

switch-off to be returned to the dc source when the transistor At low speed, the pull-out torque is roughly equal to the is turned off, rather than being dissipated in the winding or

average value of the positive half-cycle of the T–θ waveforms of the freewheeling circuits. This circuit is by far the most effi- Fig. 34.82. At high speed, the finite but fixed rise and fall times cient, and at the same time gives the fastest possible rise and of the currents and the back-emf of the winding reduces the fall times for the winding currents. extent to which the windings are energized during each switch- ing period. Consequently, the pull-out torque of the motors falls as the stepping rate (speed) increases.

For operation at high speed, the stepping rate is gradually

34.8.5.2 Bipolar Drive Circuits

increased and decreased from one speed to another. With- The bipolar drive allows the motor windings to be driven out careful acceleration and deceleration to and from a high with bidirectional currents. The four-transistor bridge drive speed, the motor will not be able to follow the stepping com- circuit of Fig. 34.91, one for each winding, is the most popu- mands and will lose its synchronism with the stepping pulses lar. The circuit can cater to the required rise and fall times of

or winding excitations. The acceleration and deceleration rates the winding by properly selecting the dc supply voltage V dc , of a stepper motor are also determined largely by the pull-out the pulse-width modulator, and the current controller gains. torque characteristic.

Some hybrid and PM motors come with four windings, two Stepper motors are known to suffer from mechanically for each phase. These may be connected in series or parallel,

induced resonance and consequent mis-stepping when its depending on the torque characteristics desired. In any case, switching rate falls within certain bands, which are largely only two drive circuits of the type indicated in Fig. 34.91 are determined by the way the developed torque varies with required. time, as the motor steps. Careful selection of stepping rate is normally employed to overcome the problem. Some shaft- mounted external damping measures may also be used when the stepping rate needs to be continuously varied, such as in

34.8.5.3 Drive Circuits for Bifilar Wound Motors

the case of machine-tool profile following. Hybrid stepping motors may also come with bifilar windings, which allow the simpler unipolar drive circuits to be used. These motors have two tightly coupled windings for each