Transfer Function of Armature Controlled DC Motor


Transfer Function of Armature Controlled DC Motor

Transfer Function of Armature Controlled DC Motor:

               The speed of DC motor is directly proportional to armature voltage and inversely proportional to flux in field winding.In armature controlled DC motor the desired speed is obtained by varying the armature voltage.This speed control system is an electro-mechanical control system.We will discuss transfer function of armature controlled dc motor.

              The electrical system consists of the armature and the field circuit but for analysis purpose, only the armature circuit is considered because the field is excited by a constant voltage.The mechanical system consist of the rotating part of the motor and load connected to the shaft of the motor.The armature controlled DC motor speed control system is shown in the below figure. 

transfer-function-armature-controlled-dc-motor


Let          R= Armature resistance,
               La = Armature inductance
               Ia  =  Armature current 
               Va = Armature voltage
               eb  = Back emf
               Kt = Torque constant
               T = Torque developed by motor
               θ = Angular displacement of shaft
               J = Moment of inertia of motor and load
              B = Frictional coefficient of motor and load
             Kb= Back emf constant 

Must Read:
The equivalent circuit of armature is shown in the below figure.

transfer-function-armature-controlled-dc-motor
By Kirchoff's voltage law, we can write,  

transfer-function-armature-controlled-dc-motor

Torque of DC motor is proportional to the product of flux and current.Since flux is constant in this system, the torque is proportional to ia alone. 
transfer-function-armature-controlled-dc-motor

The mechanical system of the motor is shown in the below figure.
transfer-function-armature-controlled-dc-motor
The differential equation governing the mechanical system of motor is given by 
transfer-function-armature-controlled-dc-motor

The back emf of DC machine is proportional to speed (angular velocity) of shaft.
transfer-function-armature-controlled-dc-motor
Must Read:
The Laplace  transform of various time domain signals involved in this system are shown below.

  L{va} = Va(s);  L{eb} = Eb(s) ; L{T}= T(s) ; L{ia} = la (s) ; L{θ} = θ(s) 

The differential equations governing the armature controlled DC motor speed control system are

transfer-function-armature-controlled-dc-motor

On taking Laplace transform of the above equations with zero initial conditions we get,

                  Ia(s) Ra + La sIa(s) + Eb(s) = Va(s)                           => (1)
                  T(s) = Kt Ia(s)                                                         => (2)
                  J s θ(s)  + B s θ(s) = T(s)                                        => (3)
                  Eb(s) = Kbθ(s)                                                      => (4)

On equating the above equations (2),(3) we get
                    transfer-function-armature-controlled-dc-motor                        => (5)

Equation  (1) can be written as,

                       (Ra + sLa) Ia(s) + Eb(s) = Va(s)                            => (6)

Substituting for Eb(s) & Ia(s) from (4),(5) respectively in equation (6)
transfer-function-armature-controlled-dc-motor

The required transfer function of armature controlled dc motor  is θ(s)/Va(s)

                         transfer-function-armature-controlled-dc-motor                  => (7)
transfer-function-armature-controlled-dc-motor

Must Read:
The transfer function of armature controlled dc motor can be expressed in another standard form as shown below.From the equation (7) we get,

transfer-function-armature-controlled-dc-motor

where  La/Ra = T = Electrical time constant
    
           J/B = Tm = Mechanical time constant

Conclusion:
           In this post we have learnt Transfer Function of Armature Controlled DC Motor.You can download this article as pdf, ppt.If you have queries you can email us @ palakalaamarnath@gmail.com

Comment below if you have something to say!