Question

Electrical machine control

A motor has the following parameter: Un=110V, Ra=0.5ohm, armature inductance L=75mH, moment of inertia J=1kgm²,Total torqu due to mechanical losses Tm=0.7Nm

1)Given Io=1.8A Deduce the factor k such that T=k.I hence deduce the no load speed.

2) the machine is running without load and we couple suddenly (at t=0) a mechanical load presenting a resistive torque of Tr=13Nm. Write the differential equations linking torque to angular speed and hence the KVL equations during the transient mode of operation.

3)write a global equation linking the angular speed w(rad/s), dw/dt and di/dt and state the hypothesis that can simplify the equation.

4) calculate and sketch the transient evolution of the current stating the estimated duration of the transient state during change of load.

Answer 1

Under no load,

the torque supplied by motor T = torque supplied for meeting the losses Tm = 0.7Nm

The electrical equivalent torque T = k*I (where k is torque constant or speed constant and I is armature current)

Under no load, the current taken by the motor I₀ = 1.8A (given)

Therefore, the constant k = T/I₀ = 0.7/1.8 = 0.3889

The back emf developed by motor E_b = k*w (where w is angular speed in rad/sec)

Using KVL, the back emf of the motor can be expressed by E_b = V - I_aR_a

where V is supply voltage and Ra is the armature resistance

Supply voltage given V = Un = 110V

Armature resistance Ra = 0.5ohm

Therefore Eb = 110 - (1.8*0.5) = 110 - 0.9 = 109.1V

Now angular speed of motor under no load w = Eb/k = 109.1/0.3889 = 280.54 rad/sec

speed in RPM is calculated as follows: w= 2*pi*N/60

No load speed of motor is N = 60*w/2*pi = (60*280.54)/ (2*pi) =2679 RPM

Under load:

Mechanical load is applied Tr = 13Nm

Now the electrical equivalent torque Te has to meet the mechanical load torque Tr and torque supplying losses Tm

Te = Jd²w/dt² + Bdw/dt + Tm

where J is moment of inertia in Kg-m²

B is friction constant

w is angular speed in rad/sec

Electrical side KVL equation can be written by

V = Ia*Ra + LdIa/dt + Eb