Simplified Drive Representations

34.7.6.2 Simplified Drive Representations

and Control

Consider the block diagram of Fig. 34.75 in which the individ- R

ual elements (blocks) are represented in terms of their transfer K c v c functions in terms of the Laplace operators.

R Here, G A (s), G C (s), G L (s), H T (s), and H F (s) represent the transfer functions of the power converter plus the motor, the controller, the load, the sensor (of speed in this example), and

C R the filter following the sensor respectively. The reference input -K 3 ω

for speed and the feedback signal are connected to a summing junction of an operational amplifier through resistors R i and

FIGURE 34.73 Transient velocity feedback controller.

R f , respectively.

964 M. F. Rahman et al.

FIGURE 34.75 Block diagram of a speed-control system.

FIGURE 34.76 Simplified representation of Fig. 34.75.

FIGURE 34.77 Further simplified representation of Fig. 34.75.

The preceding system can be simplified to that shown in where T s =T 3 +T 4 +· · · etc. A dc-motor speed-control system Fig. 34.76, and further to that in Fig. 34.77.

with current and speed sensors falls in this category. For such In general, if the individual control blocks are approximated

a system there exist two dominant time constants (poles). as first-order systems and are mutually decoupled, meaning

For such a system, a proportional plus integral controller is that each block operates in a frequency band that is far outside of the form the frequency bands of all other blocks, then the foregoing

systems can be represented by a transfer function of the form ( 1 + sτ 1 )( 1 + sτ 2 )

G c (s) =

(34.82) sτ 0 ( 1 + sτ F1 )( 1 + sτ F2 )

G 1 (s) =

One optimization criterion (Kessler’s) stipulates that τ 1 ≈

( 1 + sT 1 )( 1 + sT 2 )( 1 + sT 3 )( 1 + sT 4 ) ···

T 1 ,τ 2 ≈T 2 , and τ o ≈ 2KT s . With this stipulation, the transfer function of the complete system is given by

When T 3 and T 4 are much smaller time constants than T 1

1 and T 2 , the preceding may be approximated by

G(s) =G 1 (s)G c (s) =

(34.83) 2sT s ( 1 +T s )

V (s)

G 1 (s) ≈

and

V i = (s) 1 =

( 1 + sT 1 )( 1 + sT 2 )( 1 + sT s )

+G 2 1 + 2sT s + 2s T s

34 Motor Drives 965

For this system, Kessler’s optimization criterion stipulates that

8KT s v(t)

τ 1 = 4T s ,τ 2 =T 2 , and τ 0 = T 1

The transfer function of the complete system is then

FIGURE 34.78 Response of the optimized system of Fig. 34.75. The peak overshoot of this system to a unit step unit is usu- ally unacceptable, as indicated by the response of Fig. 34.79. This overshoot is usually reduced by inserting a first-order fil-

Note that the two filter time constants τ F1 and τ F2 are ter in the reference circuit. The filter network and the responses included in G c (s) for the sake of its realizability. These can are given in Fig. 34.80.

be relegated to frequencies far higher than the range of inter- est and can be ignored for further analysis of the system. For an unit step input of V i , the output V is given by

34.8 Stepper Motor Drives

√ v(t ) =1− 2e −1/2T s

34.8.1 Introduction

sin

for t ≥ 0. (34.85)

2T s

4 A stepper motor is a positioning device that increments its shaft position in direct proportion to the number of current

A typical output is sketched Fig. 34.78. pulses supplied to its windings. A digital positioning system If the transfer function G 1 (s) has one dominant time con- without any position or speed feedback is thus easily imple-

stant T 1 (s), as for the field current control of a dc motor, a mented at a much lower cost than with the other types of suitable controller is the form

motors, simply by delivering a counted number of switching signals to the motor. Typically, a 200 steps-per-revolution step-

G c (s) = (34.86) per motor with 5% stepping accuracy will be equivalent to a

( 1 + sT 1 )

sτ 0 ( 1 + sT F )

dc motor with a 12-bit (or 4000 counts/rev) encoder plus the closed-loop speed and position controllers for obtaining simi-

In some cases, the transfer function G 1 (s) is of the form

lar positioning resolution. This advantage, however, is obtained at a cost of increased complexity of the drive circuits. A dis-

advantage of the motor is perhaps its inability to reach an

absolute position, since the final position reached is only rel- ative to its arbitrary initial position. Nevertheless, the true

where T s is the sum of a number of short time constants, digital nature of this motor makes it a very suitable candi- associated with sensors, switching frequency, and so on. The date for digital positioning systems in many manufacturing, current controller of the dc motor with back-emf has such a automation, and indexing systems. characteristic. A suitable PI controller for this system is

The working principle of stepper motors is based on the tendency of the rotor to align with the position where the

( 1 + sτ 1 )( 1 + sτ 2 )

stator flux becomes maximum (i.e. seeking of the minimum-

reluctance position, also called the detent position). The rotor

FIGURE 34.79 Block diagram representation of a typical current controller.

966 M. F. Rahman et al.

Without input filter

0.43 A A C C

With input filter

B B 3.1T s 7.6T s

A A FIGURE 34.80 Optimized response of the system of Fig. 34.80.

Time, msec

and the stator are both toothed structures, and the stator nor- FIGURE 34.81 Cross section of a single-stack variable-reluctance mally has more than two windings to step the rotor in the stepper motor. desired direction when they are energized in certain combina- tions with current. Some motors additionally have permanent magnets embedded in the rotor that accentuate an already existing, zero-excitation detent torque. These motors hold where N is the number of phases in the stator and P is the their positions even when the stator excitations are removed number of poles in the stator. Single stack motors typically completely, a feature desirable for some applications.

has larger step angles than other types because of limitations In addition to the point-to-point stepping action, these of space for the windings. The step angle of these motor tend motors can also be operated at high slewing speed, simply to be larger than the multistack and hybrid stepper motors. by increasing the pulsing rate of phase currents. Since the

For each excited winding, the motor develops a torque angle motor is inherently a synchronous actuator, the pulsing rate (T–θ) characteristic as indicated in Fig. 34.82. Note that there has to be increased and decreased properly, so that the rotor are two equilibrium positions of the rotor, namely, X and Y , may follow it. At the end of a complete run, the motor always where the motor develops zero torque. stops at the desired incremental position or angle without any

The position X is referred to as the stable detent position, accumulated error. The only error that may be encountered is around which the rotor develops a restoring torque when dis- mainly due to the machining accuracy of the teeth in the stator placed. The restoring torque increases as the rotor is moved and rotor. This error is of the order of about 5% of one step from its detent position, becoming a maximum T max on either position/angle and it is nonaccumulative.

side of this position. The slope of the T–θ characteristic around this detent position and the maximum torque, both of which

34.8.2 Motor Types and Characteristics

depend on the level of excitation, indicate how far the rotor will be displaced under load torque. This means that the level

34.8.2.1 Single-stack Variable-reluctance

of excitation also affects the position holding accuracy of the

Stepper Motor

motor.

Single-stack motors are normally of the variable-reluctance The motor may also be excited in the sequence: AB–BC–CA type with no excitation in the rotor. The cross section of a or AB–CA–BC for forward and reverse stepping, respectively.

three-phase motor with two stator poles/phase and four rotor The two phases-on scheme develops more torque around the poles are indicated in Fig. 34.81. The motor can be stepped detent positions at the expense of twice the resistive losses. clock or anticlockwise by energizing the phase winding in the

Yet another excitation scheme is AB–B–BC–C–CA–A–AB for ABCA or ACBA sequence, respectively. The step angle, i.e. forward stepping and AB–A–AC–C–BC–B–AB for reverse step-

the angle moved by the rotor for each change in excitation ping. In this scheme, the step size is halved as opposed to the sequence, of the motor is given by

full-step size of the previous sequences. Two different levels of torque is produced for alternate detent positions. However,

degrees

(34.90) the reduced step size and the more damped nature of each step

NP

may outweigh this disadvantage.

34 Motor Drives 967

T, Nm T max

dT

Unstable detent

Stable detent

q, rad

−T max

FIGURE 34.82 Static torque characteristic of a stepper motor.

34.8.2.2 Multi Stack Variable-Reluctance

the stator excitation. The stator and rotor teeth in the other sec-

tions are not aligned. By changing the combination of excited In a multi stack variable-reluctance motor, the stator windings phases to the next in sequence, the rotor is made to move by

Stepping Motor

are stacked along the shaft. Each stack section now has the one step angle. same number of poles in the stator and the rotor. Normally each stator stack is staggered with respect to its neighbor by one/Nth of a pole pitch, where N is the number of stator/rotor

34.8.2.3 Hybrid Stepping Motor

A hybrid stepper motor has an axially oriented permanent internal details of a six-phase multi stack motor, in which each magnet sandwiched between two sections of the stator and

phases or sections. The cut out view of Fig. 34.83 shows some

stack has a phase winding between two rings, each with 32 rotor, as indicated in Fig. 34.84. The magnetic flux distributes stator and rotor poles. The step size of this motor is

radially through the two stator and rotor sections, both of

which are toothed, and axially through the back iron of the 360

(34.91) stator and the shaft. The stator has two phase windings, each Np

of which creates alternate polarities of magnetic poles in both sections of the stator. Stator windings are excited with bipolar

The excitation sequence of this motor is similar to the ones mentioned in Section 34.8.2.1, except that more excitation sequences are available. When a stator winding is energized, the

rotor poles of that section tend to align with those defined by Stator and

Rotor sections X Y

Stator winding

Rotor magnet

Stator winding FIGURE 34.83 Cut out view of a six-phase, multi stack, variable-

reluctance stepper motor. Courtesy: Pratt Hydraulics, UK. FIGURE 34.84 Axial section of the hybrid motor.

968 M. F. Rahman et al.

Phase A

Phase A

Phase B

Phase B

(a)

(b)

FIGURE 34.85 Cross section of the hybrid motor: (a) section X and (b) section Y.

currents, as opposed to the unipolar currents in the variable- reluctance motors of the two preceding sections. The magnetic flux produced by the stator windings is circumferential in each stator and rotor section, but also crosses the airgap radially. It does not, however, pass through the rotor magnet. The two rotor sections are offset by half its tooth pitch.

The rotor magnet causes to the stator and rotor teeth to settle at the minimum reluctance position with a modest amount of detent torque to keep the rotor in position, when the stator windings are not energized. The rotor magnetic flux distributes outward through stator poles 3 and 7 in section X and inward through poles 1 and 5 in section Y , as shown in Figs. 34.85a and b. When the stator windings A and B (indi- cated as dark and faint shaded, respectively) are energized with FIGURE 34.86 Rotor of a PM stepper motor. Courtesy: Escap Motors. positive and negative currents, respectively, the resulting sta- tor flux also distributes through these same poles, so that the rotor then develops a much higher detent torque (T–θ) char- setup alternate poles when energized, just as in the case of acteristic. The motor can be stepped forward or backward the hybrid motor. The rotor consists of permanent magnets, by energizing windings in sequence A ¯B − AB − ¯AB − ¯A ¯B or alternately polarized, attached to the surface of a nonmagnetic

− ¯A ¯B − ¯AB −A ¯B respectively, where the over bar indicates A ¯B disk, as shown in Fig. 34.86. The stator and rotor fluxes cross the polarity of currents in phases A and B.

the airgap, one on either side of the disk, axially. The stepping angle of a hybrid stepper motor is given by