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 v01t2 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 k1Y1s2, and the linear differential transformer, rectifier, and amplifier may be represented by E01s2 = -k2Y1s2.

Therefore, the measure of the tension is described by the relation 2T1s2 = k1Y1s2, where Y1s2 is the deviation from the equilibrium condition, and T1s2 is the vertical component of the deviation in tension from the equilibrium condition. The time constant of the motor is t = La>Ra, and the linear velocity of the wind-up roll is twice the angular velocity of the motor, that is, V01s2 = 2v01s2. The equation of the motor is then 

Wind-off roll v(1) y(t) I_ Rectifier Wind-up vo(1) roll Motor Motor FIGURE P4.10 Paper tension Linear differential transformer Amplifier eo(t) control. Ra La wo(t)