## 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.

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

The equivalent circuit of field is shown in the below figure.
By Kirchoff 's voltage law, we can write

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.
The differential equation governing the mechanical system of the motor is given by,

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

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,
=> (4)
The equation (1) can be written as

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