For a paper processing plant, it is important to maintain a constant tension on the continuous sheet of paper between the wind-off and wind-up rolls. The tension varies as the widths of the rolls change, and an adjustment in the take-up motor speed is necessary, as shown in Figure P4.10. If the wind-up motor speed is uncontrolled, as the paper transfers from the wind-off roll to the wind-up roll, the velocity v0 decreases and the tension of the paper drops [10, 14]. The three-roller and spring combination provides a measure of the tension of the paper. The spring force is equal to k1y, and the linear differential transformer, rectifier, and amplifier may be represented by e0 = -k2y.
Figure P4.10
Paper tension control.

Therefore, the measure of the tension is described by the relation 2T(s) = k1y, where y is the deviation from the equilibrium condition, and T(s) is the vertical component of the deviation in tension from the equilibrium condition. The time constant of the motor is Ñ‚ = La/Ra, and the linear velocity of the wind-up roll is twice the angular velocity of the motor, that is, v0(t) = 2w0(t). The equation of the motor is then

where ΔT = a tension disturbance.
(a) Draw the closed-loop block diagram for the system, including the disturbance ΔT(s).
(b) Add the effect of a disturbance in the wind-off roll velocity ΔV1(s) to the block diagram. (c) Determine the sensitivity of the system to the motor constant Km.
(d) Determine the steady-state error in the tension when a step disturbance in the input velocity, ΔV1(s) = A/s, occurs.

Wind-off v Wind-up roll roll Rectifier Motor Motor Linear differential transformer Amplifier le" - W -YYY Ra Lu