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文献出处：Firoozian R. Stepping Servo Motors [J] Servo Motors and Industrial Control Theory. Springer International Publishing, 2015，3（2）: 91-99.

Stepping Servo Motors

Riazollah Firoozian

A stepping motor is defined as a rotary device whose output shaft moves in discrete steps when excited from a switched DC supply. Stepping motors are very practical devices for converting digital pulse inputs into analogue shaft-output (or rotary) motion as required in modern electric or electronic equipment. Each shaft revolution can be expressed in terms of a number of discrete identical steps or increments. Each step can be triggered by a single pulse. Stepping motors can be made with a rotor made of permanent magnets or DC energized. In the latter case, brushes must be used to energize the rotor. The principal operation of both types are shown in Fig. 5.1

1 Principal Operation

A stepping motor is defined as a rotary device whose output shaft moves in discrete steps when excited from a switched DC supply. Stepping motors are very practical devices for converting digital pulse inputs into analogue shaft-output (or rotary) motion as required in modern electric or electronic equipment. Each shaft revolution can be expressed in terms of a number of discrete identical steps or increments. Each step can be triggered by a single pulse. Stepping motors can be made with a rotor made of permanent magnets or DC energized. In the latter case, brushes must be used to energize the rotor. The principal operation of both types are shown in Fig. 5.1.

Principal operation of a permanent magnet rotor stepping motor

Figure 5 shows that when the phase A–A′ are energized, the rotor remains 0?at the position shown in the figure. When the phase A–A′ are de-energized and the phase B–B′ are energized, the rotor moves 90° (one step). The process may be repeated to move the rotor one more step of 90°. If the two phases are energized simultaneously, the rotor moves 45°. By proper sequence of energizing and de-energizing steps of 45 or 90° may be achieved.

The rotor may be designed so that magnetic poles can be achieved from external power supply. This is shown in Fig. 5.2.

Stepping motors with rotor energized externally must have brushes to supply power to a

rotating rotor. Brushes need to be checked periodically for proper operation. For this reason, in servo control system permanent magnet rotor is preferred .

The stator of the stepping angles may be made with two, three, or four phases. This is shown in Fig. 5.3

Figure 5.3 shows that by properly energizing and de-energizing the phases, step angle of 60°is achieved. The rotor can be just laminated steels or permanent magnet. If the rotor is made of permanent magnet, a larger output torque can be achieved.

2 Stepping Motors with Small Step Angle

If the rotor and stator of stepping motors are made of laminated steels with many teeth, a very small angle can be achieved. A step angle of 1.8 or 0.9° stepping motors are readily available in the market. The permanent magnet is buried in the rotor. This is shown in Fig. 5.4

A typical stepping motors with small step size

By properly energizing and de-energizing the stator windings, the rotor moves to the required position with the defined phase angle. In addition, if a gear box is attached to the stepping motor, a very high accurate position can be achieved. To move the stepping motor to the required position greater than the step angle, an electronics power unit is usually attached to the motor in which a single pulse represents one step movement of the rotor. This means that correct phases are energized and de-energized. The second input pulse represents another step movement. This requires change of phases energized and de-energized .

A very simple diagram with uses of transistors to switch the power to the correct phases is shown in Fig. 5.5. It can be seen that at least four transistors are needed to perform the correct switching actions. An electronic device will send a current to the required transistors. This makes them connect to a voltage to the required phases as shown by A, B, C, and D.

3 Torque–Displacement Characteristic of a Stepping Motor

When the teeth of one energized phase of stator with the teeth of the rotor are aligned, the output torque is zero. It means that the rotor does not show any resistance to the applied output torque. When the rotor is deflected by a small amount, an attracting pulling force develops between the teeth of the rotor and stator which shows itself as a torque on the rotor . The behavior can be simulated by a sinusoidal output torque. This is shown in Fig. 5.6, when two phases are energized.

Torque–displacement characteristic of generalized stepping motor

The relationship between the output torque and rotor displacement can be approximated as Tm:=nK1t?I?sin( θ )

(5.1)

In equation (5.1), n is number of phases energized, T m is the developed torque, K1t is motor torque constant , I is the current which may be written as

I:=NRT??

(5.2)

where N RT, φ are numbers of rotor teeth and step angle. For a specific motor, n,K1t are constants and Eq. (5.1) may be written as

Tm:=K2t?I?sin( θ )

(5.3)

where K2t is constant and I is the current flowing in the stator windings and θ is the rotor position. Equation (5.3) is nonlinear and to study the dynamic behavior of stepping motors , it must be linearized as

Tm:=Kt?I?Kθ?θ

(5.4)

where

Kt:=ddITm θ :=constantKt:=K2t?sin( θ )

(5.5)

K θ :=dd θTmI:=constantKθ:=K2t?I?cos( θ )

(5.6)

The second term in Eq. (5.4) is negative because as θ increases over one step of stepping motor, the torque reduces . When θ moves one step, the torque becomes zero. The torque constant (K t) and the step angle are given by manufacturers. Therefore, K θ can easily be evaluated .

4 Dynamic Response Characteristic over One Step Movement

It is important first to study the dynamic response of stepping motors for single step movement. To do this the linearized Eq. (5.4) must be used. In addition several assumptions have to be made. For the simplest model, the inductance of the windings is ignored. The voltage equation simply becomes

V:=RI

(5.7)

Eliminating the variable I from Eqs. (5.4) and (5.7) gives

Tm:=KtR?V?Kθ?θ

(5.8)

The equation of motion for the rotor assuming the total inertia of J and a viscous mechanical damping of C may be written as

Tm:=Js2 θ +Cs θ +TI

(5.9)

where T l is the external torque applied to the motor. Eliminating the variable T m from the two Eqs (5.8) and (5.9) and with some algebraic manipulations, the transfer function becomes θ:=KtRKθ?V?T1JKθ?s2+CKθs+θ

(5.10)

It can be seen that the governing transfer function in simplest form is a second order with two input variables of voltage and external torque. The coefficients of the characteristic equation gives the natural frequency and the damping ratio as

1ω2n:=JKθωn:=KθJ???√

(5.11)

The natural frequency gives the speed of response and it can be seen that as the inertia increases the natural frequency reduces and as K θ increases the natural frequency increases. The damping ratio is

2ξωn:=CKθξ:=C2?1Kθ?J?????√

(5.12)

If the mechanical damping coefficient C is zero, the rotor will oscillate continuously. In practice, there is some mechanical friction that increases the damping ratio. It means that the rotor moves one step and will oscillate little and the teeth become aligned with the stator. Both K θ and J reduce the damping ratio .

For a better mathematical model, the inductance of the stator windings must be taken into account. This only changes the voltage equation and the transfer function becomes third order which means that the motor might become unstable.（完整文献请到百度文库）