Modeling Approach
This section presents the modeling and simulation for an 8-pole (four-phase), PM stepper motor
with a resolution of 1.80 per step and a 0 to 1000 step per second (sps) speed range. The
motor is
directly attached to a load having a total inertia value (including the rotor mass) Jm kg_m2 and a total
viscous damping factor Bm N·m/rad/sec. The motor is driven by a four-phase driver that produces pulses
to each of the four phases sequentially.
The dynamic performance of the stepper motor system (drive, motor, and load) is simulated. A
four-phase, 1.80 PM stepper motor has 8 stator poles with 2 or more teeth per pole and a 50-tooth rotor.
Each pole has one winding that produces a magnetic flux into or out of the rotor, depending on the
direction of the current flow.
The stator-rotor configuration is presented in Figure1.
Figure 1. Stepper Motor Configuration
The four pole pairs (phases) are labeled A, B, C, D. From Figure 1, it can be seen that clockwise
phase excitation (A BCD) results in counterclockwise rotor motion, and counterclockwise phase
excitation (A DCB) in clockwise rotor motion. Because all four phases are identical, the electromagnetic
torque produced by one phase is first modeled.
The total electromagnetic torque produced by the four phases is obtained by copying the one-phase
model three times and summing the four individual phase torques.
The electromagnetic torque produced by the motor is applied directly to the load with no gear
reduction present. The load is modeled as a lumped inertia-damper that includes the motor
contributions as well as those of the load. The load model is forced by the difference between the
applied electromagnetic torque from the motor and the reaction torque from the load. The load model
produces two outputs, rotor speed and rotor angle, which are fed back and used it the motor model.
The drive circuit is modeled as a pulse generator with four sequentially triggered phases such that
only one phase is on at any given time. The drive circuit model assumes ideal switching between
phases and does not model the L/R time constants or the transistor switching behavior the model is
suitable for use in the present application; however, more detail could be easily: included if required.
The drive circuit has two command inputs, a step per second command sps, and a direction command,
dir. It produces four voltage outputs, one for each phase of the motor.
The top level block diagram of the stepper motor system is presented in Figure 2. The system
consists of three components: the drive, the stepper motor, and the load.
Figure 2. Stepper Motor System Top Level Block Diagram
The sps command is selectable in the 0 to 1000-sps range. The dir command is also selectable and
has two states, +1 or -1, where +1 forces clockwise rotor rotation and -1 force counter clockwise
rotation.
The digital motion control of the stepper motor requires that the number and the frequency of
pulses be calculated by the computer and sent to the stepper motor to produce the required motion.
Drive Equations and Block Diagram Model
For specified steps per second, one phase of the four-phase drive model produces on voltage pulse
at a rate of sps/4 times per second. The division by four accounts for the numb of phases. The duration
of time that the pulse is "on" is l/sps seconds.
For example, the phase voltage corresponding to an sps =8 of phase A during a l-second time span is igh
between times 0 and 0.125 seconds and between 0.5 and 0.625 seconds. The phase voltages for phases
B, C, and D are identical in shape but are delayed by 1/sps, 2/sps and 3/sps seconds, respectively.
Figure 3 presents the block diagram used to model the drive circuit behavior.
Figure 3. The drive circuit model
The drive model produces positive-valued sequential pulses that will move the rotor in one
direction. To achieve bidirectional movement, the phase voltage signal, Vx is multiplied by the direction
reference, dir.
Motor Equations and Block Diagram Model
The PM motor consists of four identical phases allowing the motor model to be developed based on
a model of one phase, which is then triplicated for the remaining three phases. The one-phase model
operates as follows.
As a voltage pulse occurs from the drive circuit, the stator winding produces a current due to the
difference between the voltage pulse and the back-emf voltage. Neglecting the mutual inductance, the
winding is modeled as a self-inductance, due to changes in the phase current, and a resistance.
The resulting phase current model is represented by Equation 1. For brevity, the time dependence has
been dropped on the current and voltage signals.
1
-----------------------(1)
i=
−
R + LD vx vbemf
(
)
Where D is the differential operator
i is the phase current
R is the phase resistance
L is the phase inductance
Vx is the driver voltage and
Vbemf is the back e.m.f. voltage
The rotor motion creates a flux linkage in the windings. This causes a back-emf voltage, which is
proportional to the rotor speed and varies periodically with the rotor position according to Equation 2.
.
v
bemf
= − K bemf θ ' sin (r∆θ ) ---------------(2)
Kbemf = back emf constant
θ’ = rotor speed in rad / sec
r = no of rotor teeth
∆θ = delta rotor angle in radians, ranging from 0 to step angle
The self-inductance, L, used previously also varies with the delta rotor position; The variation
is periodic and is represented by Equation 3.
L = L1 + L2 cos(r∆θ ) ---------------(3)
Much as in a DC motor, torque in a PM stepper motor is proportional to the phase current by a
torque constant due to the constant flux from the permanent magnet. However, it differs' in its
dependence on the flux produced by the phase current, which varies periodically with the rotor
position.
Equation 4 presents the electromagnetic torque equation for the PM stepper motor.
T e = −k.i.sin (r∆θ ) -------------(4)
Te = electromagnetic torque
K = torque constant
i = phase current
The complete block diagram model for the four-phase PM motor is presented in Figure 4. The
contents of the phase B, C, and D blocks are identical to that of the phase A model.
Figure 4. Block diagram model of 4 phase stepper motor
Figure 5 illustrates the angular motion of the rotor as it travels over a 1.80 interval. The ringing
effect, a common feature of the stepper motor response, can sometimes be attenuated electrically or
by the load. However, it is difficult to remove completely. Therefore, when applying a stepper
motor actuator, you should expect this ringing behavior and factor it into the system design.