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

high
torque,

speed
controllability over a wide range,

portability,

wellbehaved
speedtorque characteristics,

and
adaptability to various types of control methods,
DC motors are widely used in numerous control applications,
including
 robotic
manipulators,
 tape
transport mechanisms,
 disk
drives,
 machine
tools,
 and
servovalve actuators.
The transfer function of the DC motor will be developed for
 a
linear approximation to an actual motor,
 and
secondorder effects, such as hysteresis and the voltage
drop across the brushes, will be neglected.

The input voltage may be applied to the field
or armature
terminals.

The
airgap 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
i_{a}=I_{a} is a constant
armature current, and k_{m}
is defined as the motor
constant. The field current is related
to field voltage as
The
motor torque T_{m}(s)
is equal to torque delivered to the load. This relation may
be expressed as
Where
T_{L}(s)
is the load
torque and
T_{d}(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 windgust 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 motorload combination,
with T_{d}(s)=0,
is:
The
block diagram model of the fieldcontrolled 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 τ_{f}=L_{f}/R_{f}
and τ_{L}=J/b.
Typically, one finds that τ_{L}>τ_{f}
and often the field time constant may be neglected.
Armaturecontrolled
DC motor
The
armaturecontrolled DC motor uses the armature
current i_{a}
as control
variable.
The
stator field can be established by
 a
field coil and current or
 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
k_{m}
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
V_{b}(s)
is the back
electromotiveforce 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 armaturecontrolled 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 steadystate 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 steadystate 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.
