Transfer Function of Field Controlled DC Motor


Transfer Function of Field Controlled DC Motor

Transfer Function of Field Controlled DC Motor:

        The speed of a DC motor is directly proportional to armature voltage and inversely proportional to flux.In field controlled DC motor the armature voltage is kept constant and the speed is varied by varying the flux of the machine.Since flux is directly proportional to field current, the flux is varied by varying field current.Here we will learn derivation of transfer function of field controlled dc motor.

       The speed control system is an electro-mechanical control system.The electrical system consists of armature and field circuit but for analysis purpose, only field circuit is considered because the armature is excited by a constant voltage.The mechanical system consists of the rotating part of the motor and the load connected to the shaft of the motor.The field controlled DC motor speed control system is shown in the below figure.For this field controlled DC motor we shall find transfer function.
transfer-function-field-controlled-dc-motor


Let              Rf = Field resistance 
                   Lf = Field inductance 
                   if = Field current 
                   Vf= Field voltage
                   T = Torque developed by motor 
                  K= Torque constant
                   J = Moment of inertia of rotor and load
                   B = Frictional coefficient of rotor and load

Must Read:
The equivalent circuit of field is shown in the below figure.
transfer-function-field-controlled-dc-motor
By Kirchoff 's voltage law, we can write
transfer-function-field-controlled-dc-motor

The torque of DC motor is proportional to product of flux and armature current. Since armature current is constant in this system, the torque is proportional to flux alone, but flux is proportional to field current.

                                                T ∝ ir
                                        Torque , T = Ktf ir

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

The Laplace transform of various time domain signals involved in this system are shown below.

            L{if} = If(s) ; L{T} = T(s) ; L{vf} = Vf(s) ; L{θ} = θ(s) 

The differential equations governing the field controlled DC motor are,in the derivation of transfer function of field controlled dc motor

transfer-function-field-controlled-dc-motor
Must Read:
On taking Laplace transform of the above equations with zero initial condition we get,

                                 Rf If(s) + Lf s If(s) = Vf(s)                             => (1)
                                          T(s) = Ktf If(s)                                     => (2)
                                     J s² θ(s) + B s (s) = T(s)                            => (3)

Equating equations (2) & (3) we get,

transfer-function-field-controlled-dc-motor
                                                                                                      => (4)
The equation (1) can be written as

                                  (Rf + sLf) If(s) = Vf(s)                                     => (5)

Must Read:
On substituting to If(s) from equation (4) in equation (5) we get,transfer function of field controlled dc motor.

transfer-function-field-controlled-dc-motor

where            Km = Ktf/Rf B = Motor gain constant
                      Tf  = Lf/Rf    = Field time constant
                       Tm = J/B     = Mechanical time constant

Conclusion:
              In this post we have learnt Transfer Function of Field Controlled DC Motor.You can download this article as pdf,ppt.If you have queries you can mail us @ palakalaamarnath@gmail.com.Comment below if you have any queries!