TORQUE EQUATIONS FOR ROTATING SYSTEMS The Newtons Law states that, the net force acting on a body of mass M equals to the rate of change of its mechanical momentum, which is the product of its mass and its velocity in the direction of the net force. In the equation form, this is given byF= = (1)Where F is the net force acting on the body, M is the mass of the body and v is its velocity. This is illustrated by Figure 1.

Figure 1: Translational motionWith constant mass, (1) can be written asF= (2)For rotational motion (which is the case for rotating electrical machines), the force, the mass and the linear velocity is equivalent to the torque, the moment of inertia and the angular velocity, respectively. Equation (1) can therefore be written as T = (3)Where T is the net torque, J is the moment of inertia and is the angular velocity. The rotational system which is analogous to the translational system of Figure 1 is shown in Figure 2.

Figure 2: Rotational motionFor most of the cases, J is constant thus reducing (3) toT = J (4)In terms of the angular position, , this can be written asT = J (5)For rotating electrical machines, the net torque is given byT = Te-Ti = T = J (6)Where Te is the internal electrical torque produced by the motor, Tl is the load torque and/or the internal friction of the motor. T is the available torque at the shaft and is responsible for accelerating the inertia of the motor. T is also known as the dynamic torque and it only exists during the transient (i.e. acceleration and deceleration). In order to accelerate in forward direction, Te Tl must be positive; which means that the applied electrical torque must be larger than the load torque. In order to decelerate, the net torque must be negative; the electrical torque must be made smaller than the load torque and the motor operates in braking mode more on this later. Note that the speed is always continuous. A discontinuity in speed (i.e. step change in speed) theoretically will require an infinite torque. This is analogous to the voltage and current across a capacitor in which discontinuity in capacitor voltage is not allowed as it correspond to an infinite capacitor current. Equation (4) relates the torque and the mechanical speed (or position) of the machine. For a given electrical torque profile, with the known moment of inertia and the load torque, the speed profile of the drive system can be determined. In a torque-controlled drive system, the speed is governed by the load. If the load torque comprise of only the frictional torque which is proportional to the speed, (4) can be written asTe= (7)

Usually in a cascaded closed-loop control system in which the speed is to be controlled, the reference torque will be generated by the speed controller. In such cases, the torque will be governed by the speed. If we multiply (7) with the angular speed, we obtain an equation describing the power balance,(8)COMPONENTS OF LOAD TORQUE, TL In general, the load torque Tl can be classified into two types: the passive load torque (frictional torque) and the active load torque. Frictional toque exists only when there is motion and it always opposes the driving torque. Active load torque on the other hand, is independent of the direction of motion.Frictional Torque Moving parts of the motor and load constitute the frictional torque. There are several types of frictional as described in Figure 1 and explained below: Coulomb friction exists in bearings, gears, coupling and brakes. It is almost independent of speed. Viscous friction exist in lubricated bearings due to the laminar flow of the lubricant. It is directly proportional to the speed. Windage friction occurs due the turbulent flow of air or liquid. It is directly proportional to the square of speed.

Figure XX: frictional torqueIn practical drive system consisting of load and motor, all components of friction described above exist simultaneously. However, in most of the cases, only one or two components are dominating. For instance, a fan or a propeller will typically have the windage friction dominating, whereas in paper mill and machine tools, the dominating one could be the viscous friction.Constant torqueThe direction of constant load torque is independent of speed it retains the direction even when the direction of rotation reverses or changes, e.g. gravity, tension or compression undergone by elastic body. This type of torque is capable of driving the motor under equilibrium and is said to be an active torque.

Figure YY: Constant load torque: gravitational force

QUADRANT OPERATION OF A DRIVE SYSTEMThe T plane with motors shaft cross sectional area is shown in figure 4.The positive or forward speed is arbitrarily chosen in counterclockwise direction (it can also be chosen as clockwise). The positive torque is in the direction that will produce acceleration in forward speed, as shown in figure ZZ.The plane is divided into 4 quadrants, thus 4 modes of operation. The quadrants are marked as I, II, III and IV.Quadrant I:Both torque and speed are positive the motor rotates in forward direction, which is in the same direction as the motor torque. The power of the motor is the product of the speed and torque (P= Te), therefore the power of the motor is positive. Energy is converted from electrical form to mechanical form, which is used to rotate the motor. The mode of operation is known as forward motoring.

Figure ZZ: four quadrant operation of motorQuadrant II: The speed is in forward direction but the motor torque is in opposite direction or negative value. The torque produced by the motor is used to break the forward rotation of the motor. The mechanical energy during the braking is converted to electrical energy thus the flow of energy is from the mechanical system to the electrical system. The product of the torque and speed is negative thus the power is negative, implying that the motor operates in braking mode. The mode of operation is known as forward braking. Quadrant III:The speed and the torque of the motor are in the same direction but are both negative. The reverse electrical torque is used to rotate the motor in reverse direction. The power, i.e. the product of the torque and speed, is positive implying that the motor operates in motoring mode. The energy is converted from electrical form to mechanical form. This mode of operation is known as reverse motoring.Quadrant IV:The speed is in reverse direction but the torque is positive. The motor torque is used to break the reverse rotation of the motor. The mechanical energy gained during the braking is converted to electrical form thus power flow from the mechanical system to the electrical system. The product of the speed and torque is negative implying that the motor operates in braking mode. This mode of operation is known as reverse braking.