Consider the elementary motor-speed regulator scheme shown in Figure P16.2.23 for a separately…

Consider the elementary motor-speed regulator scheme shown in Figure P16.2.23 for a separately excited dc motor, whose armature is supplied from a solid-state controlled rectifier. The motor speed is measured by means of a dc tachometer generator, and its voltage et is compared with a reference voltage ER. The error voltage ER −et is amplified and made to control the output voltage of the power-conversion equipment, so as to maintain substantially constant speed at the value set by the reference voltage. Let the armature-circuit parameters be Ra and La, and the speed–voltage constant of the motor be Km, with units of V·s/rad. Assume that the combination of A and P is equivalent to a linear controlled voltage source vs = KA (error voltage), with negligible time lag and gain KA. Assume also that the load torque TL is independent of the speed, with zero damping. Neglect no-load rotational losses.

(a) Develop the block diagram for the feedback speed-control system with ER/Kt, the steady state no-load speed setting, as input, and Ωm as output. Kt is the tachometer speed–voltage constant in V/(r/min).

(b) With TL = 0, evaluate the transfer function Ωm/ER.

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(c) With ER = 0, obtain the transfer function Ωm/TL.

(d) Find the expressions for the under damped natural frequency ωn, the damping factor α, and the damping ratio ξ = α/ωn.

(e) For a step input ∆ER, obtain the final steady state response ∆ωm(∞), i.e., evaluate ∆ωm(∞)/∆ER.

(f) Evaluate ∆ωm(∞)/∆TL for the step input ∆T=of a load torque.

 

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