function of the DC motor
The DC motor is a power actuator device that delivers energy
to a load, as shown in figure below.
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
controllability over a wide range,
adaptability to various types of control methods,
DC motors are widely used in numerous control applications,
The transfer function of the DC motor will be developed for
linear approximation to an actual motor,
second-order effects, such as hysteresis and the voltage
drop across the brushes, will be neglected.
The input voltage may be applied to the field
air-gap flux Φ
of the motor is proportional to the field
current, provided the field is unsaturated.
torque developed by motor is assumed to be related linearly
and the armature current as follows:
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.
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,
ia=Ia is a constant
armature current, and km
is defined as the motor
constant. The field current is related
to field voltage as
motor torque Tm(s)
is equal to torque delivered to the load. This relation may
be expressed as
is the load
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
by rearranging equation, we have
Therefore, the transfer function of the motor-load combination,
block diagram model of the field-controlled DC motor is shown
in figure below.
the transfer function may be written in terms of the time
constants of the motor as
Typically, one finds that τL>τf
and often the field time constant may be neglected.
armature-controlled DC motor uses the armature
stator field can be established by
field coil and current or
a constant field current is established in a field coil, the
motor torque is
a permanent magnet is used, we have
is a function of the permeability
of the magnetic material.
The armature current is related to the input voltage applied
to the armature by
is the back
electromotive-force voltage proportional
to the motor speed.
Therefore, we have
is the transform of the angular speed and the armature current
So, the load torque is
relations for the armature-controlled DC motor are shown schematically
in figure below.
Td(s)=0, we solve to obtain the transfer function
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.