An armature-controlled dc motor is sometimes used in speed and position control systems. The dc motor operation is described by the following equations:
where:
e = armature voltage, V
i = armature current, A
ω = motor speed, rad/s
q = motor torque, N.m
J = moment of inertia of the load, kg.m2
b = damping resistance of the load, N.m/(rad/s)
R = armature resistance, Ω
L = armature inductance, H
Ke = back emf constant of the motor, V/(rad/s)
Kt = torque constant of the motor, N.m/A
A small permanent-magnet dc motor has the following parameter values:
J = 8 x 10
-4 kg.m
2
b = 3 x 10-4 N.m/(rad/s)
R = 1.2Ω
L = 0.020 H
Ke = 5 x 10-2 V/(rad/s)
Kt = 0.043 N.m/A
Substitute these parameters into the preceeding equations to obtain the exact differential equations of the dc motor. Determine the transfer function, Ω(s)/E(s), by transforming all three equations into frequency-domain algebraic equations. Use algebraic operations to obtain the ratio of Ω/E, which is the desired transfer function.
Here is what I have so far, then I got stuck:
Substitute the parameters into the 3 equations:
e = 12i + 0.020 di/dt + 5 x 10-2 r/rad/sω
i = q/0.043 N.m/A
q = J/dω/dt
The method of finding the exact differential equation is not so clear to me fromthe three equations. Should the 3 equations be resolved into 1 differential equation? Please help with a step-by-step method to solving these types of problems