Dc motors and Reduction Techniques and Transfer Function

DC motors
In control systems, a dc motor can be used in different control modes. The two
control modes used are:
 Armature control with fixed field current.
 Field control with fixed armature current.
Armature control

𝑅𝑎 𝐿𝑎 𝑖𝑓 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
+



e 𝑖𝑎 𝑒𝑏 +



_
𝜃
𝐽, ϝ




Where:
𝑅𝑎 = resistance of the armature winding
𝐿𝑎 = inductance of the armature winding
𝑖𝑎 = the armature current
𝑖𝑓 = the field current
𝑒 = the applied armature voltage
𝑒𝑏 = the back emf
𝑇 = the torque developed by the motor
𝜃 = the angular displacement of the motor shaft
𝐽 = the equivalent moment of inertia of motor and load referred to the motor shaft
ϝ = the viscous friction coefficient

, In the linear range of magnetization curve, air gap flux is proportional to the field
current.
∅ = 𝑘𝑓 𝑖𝑓
The torque developed by the motor is proportional to product of armature current
and the air-gap flux, hence we have:
𝑇 = 𝑘1 𝑘𝑓 𝑖𝑓 𝑖𝑎
For armature controlled dc motor, the field current is kept constant and hence we
have:
𝑇 = 𝑘 𝑇 𝑖𝑎 ⋯ ⋯ ⋯ ⋯ ⋯ (1)
𝑘 𝑇 = 𝑘1 𝑘𝑓 𝑖𝑓 is the motor torque constant.
The Kirchhoff’s voltage law for the armature circuit is:
𝑑𝑖𝑎
𝐿𝑎 + 𝑅𝑎 𝑖𝑎 + 𝑒𝑏 = 𝑒 ⋯ ⋯ ⋯ (2)
𝑑𝑡
The mechanical torque equation is:
𝑑2𝜃 𝑑𝜃
𝑇 =𝐽 2 +ϝ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (3)
𝑑𝑡 𝑑𝑡
The back emf (𝑒𝑏 ) is proportional to speed and is given by:
𝑑𝜃
𝑒𝑏 = 𝑘𝑏 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (4)
𝑑𝑡
Taking Laplace transform of the above equations we get:
𝑇(𝑠) = 𝑘 𝑇 𝐼𝑎 (𝑠) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (5)

𝑠𝐿𝐼𝑎(𝑠) + 𝑅𝐼𝑎(𝑠) + 𝐸𝑏(𝑠) = 𝐸(𝑠) ⋯ ⋯ ⋯ (6)

𝑇(𝑠) = 𝑠 2 𝐽𝜃(𝑠) + 𝑠ϝ𝜃(𝑠) ⋯ ⋯ ⋯ ⋯ ⋯ (7)
𝐸𝑏 = 𝑠𝑘𝑏 𝜃(𝑠) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (8)
From (6) we have:
(𝑠𝐿𝑎 + 𝑅𝑎 )𝐼𝑎(𝑠) = 𝐸(𝑠) − 𝐸𝑏(𝑠) ⋯ ⋯ (9)