Theory & Practice of Electromagnetic Design of DC Motors & Actuators
George P. Gogue & Joseph J. Stupak, Jr.
10.1 Types of Motors:
A stepper motor is a controlled rotation device in which the current to the stator coils is switched at a predetermined rate. Even though a stepper motor is basically a brushless dc motor, it allows controlled incremental rotation of that conventional motor. Switching the current to successive coils is controlled by a switching circuit. Torque can be analyzed as it is developed by one coil at a time. Its value depends on the stator and rotor fields and the angle between them. The power angle is, however, dependent on the number of steps/revolution and the type of design of which there are three:
1. Variable reluctance type:
In this type of motor, rotation results from the electromagnetic torque between the teeth of the rotor and the stator. Figure 10.1 shows how energizing one coil can cause the rotor to move half a stator slot-pitch or 15°.
Due to the low inertia of this type of motor, the acceleration time can be very small.
2. Permanent-magnet Type:
The rotor in this type consists of a permanent-magnet which is radially magnetized as shown in Figure 10.2. The magnet increases the inertia of the motor substantially but it also provides a holding torque when the winding is not energized. This provides a sharp contrast with the variable reluctance motor. The developed torque is a combination of the electromagnetic torque and reluctance torque. The step angles in this type can be made smaller than the VR type.
3. Hybrid Type:
A combination of features from the above two types is used in this design to achieve the smallest step angles and highest accuracy possible. Figure 10.3 shows a cross section of this design but does not show the axially magnetized disc magnet around the shaft.
10.2 Torque Characteristics:
Special torque terms are used in stepper motors to describe its unique behavior. Figure 10.4 shows these torque values and operating ranges vs. speed.
The detent torque is only seen in permanent magnet steppers, whereas the holding torque is developed by all types of steppers when the winding is energized. Both above values are at zero speed and usually have a ratio of approximately 1:10. The motor can develop a particular value of torque at a particular speed provided it is within the pull-out torque curve. However, to be able to develop that torque from standstill conditions, only the values within the pull-in torque curve are possible.
The curves in Figure 10.4 above describe the torque developed under continuous running conditions, or over a limited ramping cycle of acceleration and deceleration. However, the stepper motor behavior in response to a one-step command is also important. Rise time and settling time are functions of the motor and load inertias. The over-shoot is also determined in comparison with the step size of the motor. These parameters constitute the step response of a stepper motor as shown in Figure 10.5.
10.3 Electromagnetic principles:
As seen from the energy considerations of Section 6.2 for a brushless dc motor, half of the energy withdrawn from the source is used for mechanical work and half is stored in the field. This is also true in the stepper motors and Equation (6.10) applies equally well here. Reference 32 gives a detailed description of how this relationship is derived electromagnetically and also in terms of circuit parameters.
The latter yields an expression that in turn is applicable to brushless dc motors also. This expression is based on the fact that the magnetic energy can also be expressed as:
where L is coil inductance.
The force can then be shown to be:
In the relationships above, the general form is based on linear characteristics of the steel. However, in practical designs the steel is operated at levels close to saturation. Going back to Equation (6.13), it can be shown that the change in magnetic energy associated with the movement of the rotor is related to the area under the B/H curve of the material. This is true since the B/H curve is also a curve and the work W done during the rotor movement is:
Two other basic relationships can be used in this analysis:
where dx/dt is the speed of movement of the rotor. This yields the familiar relationship when the B/H characteristics are linear:
But if the steel is saturable, the amount of F becomes considerably more than 1/2 BI and very close to BI. This, however, requires a small air gap between rotor and stator, making the construction of such motors quite difficult. The torque equation for the hybrid stepper motor, which is the most popular motor, is:
The above equation for the torque developed by one phase is based on fundamental relationships. The first term is the conventional relationship between the developed power and developed torque through the speed S. The second term is specific to the variable reluctance type stepper motors in which the coil inductance varies with rotor position. The analysis behind Equation (10.1) for the magnetic energy in the air gap is the same as that for the second term in Equation (10.7).
10.4 Stepper Design Tips:
The general relationship for one or two phase operation of stepper motors is (Reference 33)
When designing the stator and rotor teeth it is useful to remember some of the general guidelines that have been established over the years (References 32 and 34).
1. Tooth width/tooth pitch ratio of about 0.5
2. Slot depth in stator of about half tooth pitch
3. Rotor slot semicircular
4. Stator slot, semicircular or rectangular
5. Air gap range 1 to 2 mils