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Transfer function of the DC motor 

Transfer function of the DC motor

The DC motor is a power actuator device that delivers energy to a load, as shown in figure below.

The DC motor converts direct current (DC) electrical energy into rotational mechanical energy. A major fraction of the torque generated in the rotor (armature) of the motor is available to drive an external load.

Because of features such as

  1. high torque,
  2. speed controllability over a wide range,
  3. portability,
  4. well-behaved speed-torque characteristics,
  5. and adaptability to various types of control methods,

DC motors are widely used in numerous control applications, including

  1. robotic manipulators,
  2. tape transport mechanisms,
  3. disk drives,
  4. machine tools,
  5. and servovalve actuators.

The transfer function of the DC motor will be developed for

  1. a linear approximation to an actual motor,
  2. and second-order effects, such as hysteresis and the voltage drop across the brushes, will be neglected.
  3. The input voltage may be applied to the field or armature terminals.
  4. The air-gap flux Φ of the motor is proportional to the field current, provided the field is unsaturated.

So that

The torque developed by motor is assumed to be related linearly to Φ and the armature current as follows:

It is clear from above equation that, to have a linear system, one current must be maintained constant while the other current becomes the input current.

 Field current controlled motor

First, we shall consider the field current controlled motor, which provides a substantial power amplification. Then we have, in Laplace transform notification,

Where ia=Ia is a constant armature current, and km is defined as the motor constant. The field current is related to field voltage as

The motor torque Tm(s) is equal to torque delivered to the load. This relation may be expressed as

Where TL(s) is the load torque and Td(s) is the disturbance torque, which is often negligible.

However, the disturbance torque must be considered in systems subjected to external forces such as antenna wind-gust forces. The load torque for rotating inertia, as shown in figure below

is written as

So, by rearranging equation, we have

Therefore, the transfer function of the motor-load combination, with Td(s)=0, is:

The block diagram model of the field-controlled DC motor is shown in figure below.

Alternatively, the transfer function may be written in terms of the time constants of the motor as

Where τf=Lf/Rf and τL=J/b. Typically, one finds that τLf and often the field time constant may be neglected.

 Armature-controlled DC motor

The armature-controlled DC motor uses the armature current ia as control variable.

The stator field can be established by

  1. a field coil and current or
  2. a permanent magnet.

When a constant field current is established in a field coil, the motor torque is

when a permanent magnet is used, we have

Where km is a function of the permeability of the magnetic material.

The armature current is related to the input voltage applied to the armature by

Where Vb(s) is the back electromotive-force voltage proportional to the motor speed. Therefore, we have

Where ω(s)=sθ(s) is the transform of the angular speed and the armature current is

So, the load torque is

 

The relations for the armature-controlled DC motor are shown schematically in figure below.

 

Let Td(s)=0, we solve to obtain the transfer function

 

However, for many DC motors, the time constant of the armature, (964)a=La/Ra, is negligible; therefore :

 

Where the equivalent time constant (964)1=Raj/(Rab+KbKm).

Note that Km is equal to Kb. This equality may be shown by considering the steady-state motor equation and the power balanced when the rotor resistance is neglected.

The power input to rotor is (Kb(969))ia, and the power delivered to the shaft is T(969). In the steady-state condition, the power input is equal to the power delivered to the shaft so that (Kb(969))ia=T(969); since T=Kmia, we find that Kb=Km.

Electric motors are used for moving loads when a rapid response is not required and for relatively low power requirements. Typical constants for a fractional horsepower motor are provided in table below.