Power
Centrifugal loads, such as fans and blowers, require torque that varies with speed2, and load power that varies with speed3. Similarly, wind turbines produce torque that is proportional to speed2.
In constant‐torque loads such as conveyors, hoists, cranes, and elevators, torque remains constant with speed, and the load power varies linearly with speed. In squared‐power loads such as compressors and rollers, the torque varies linearly with speed and the load power varies with speed2. In constant‐power loads, such as winders and unwinders, the torque beyond a certain speed range varies inversely with speed and the load power remains constant with speed.
2‐9 FOUR‐QUADRANT OPERATION
In many high‐performance systems, drives are required to operate in all four quadrants of the torque‐speed plane, as shown in Fig. 2-16b.
Fig. 2-16 (a) Electric drive and (d) four-quadrant operation.
The motor drives the load in the forward direction in quadrant 1, and in the reverse direction in quadrant 3. In both of these quadrants, the average power is positive and flows from the motor to the mechanical load. In order to control the load speed rapidly, it may be necessary to operate the system in the regenerative braking mode, where the direction of power is reversed, so that it flows from the load into the motor, and usually into the utility (through the power‐processing unit). In quadrant 2, the speed is positive, but the torque produced by the motor is negative. In quadrant 4, the speed is negative and the motor torque is positive.
2‐10 STEADY‐STATE AND DYNAMIC OPERATIONS
As discussed in Section 2‐8, each load has its own torque‐speed characteristic. For high‐performance drives, in addition to the steady‐state operation, the dynamic operation – how the operating point changes with time – is also important. The change of speed of the motor‐load combination should be accomplished rapidly and without any oscillations (which otherwise may destroy the load). This requires a good design of the closed‐loop controller, as discussed in Chapter 5, which deals with the control of drives.
2‐11 REVIEW QUESTIONS
1 1. What are the MKS units for force, torque, linear speed, angular speed, speed, and power?
2 2. What is the relationship between force, torque, and power?
3 3. Show that torque is the fundamental variable in controlling speed and position.
4 4. What is the kinetic energy stored in a moving mass and a rotating mass?
5 5. What is the mechanism for torsional resonances?
6 6. What are the various types of coupling mechanisms?
7 7. What is the optimum gear ratio to minimize the torque required from the motor for a given load‐speed profile as a function of time?
8 8. What are the torque‐speed and the power‐speed profiles for various types of loads?
REFERENCES
1 1. Bosch (1993). Automotive Handbook. Robert Bosch GmbH.
2 2. Nondahl, T. (1998). Proceedings of the NSF/EPRI‐Sponsored Faculty Workshop on “Teaching of Power Electronics” (25–28 June 1998). University of Minnesota.
FURTHER READING
1 Gross, H. (ed.) (1983). Electric Feed Drives for Machine Tools. New York: Siemens and Wiley.
2 (1980). DC Motors and Control ServoSystem – An Engineering Handbook, 5e. Hopkins, MN: Electro‐Craft Corporation.
3 Spong, M. and Vidyasagar, M. (1989). Robot Dynamics and Control. Wiley.
PROBLEMS
1 2‐1 A constant torque of 5 Nm is applied to an unloaded motor at rest at time t = 0. The motor reaches a speed of 1800 rpm in 3 s. Assuming the damping to be negligible, calculate the motor inertia.
2 2‐2 Calculate the inertia if the cylinder in Example 2-2 is hollow, with the inner radius r2 = 4 cm.
3 2‐3 A vehicle of mass 1500 kg is traveling at a speed of 50 km/h. What is the kinetic energy stored in its mass? Calculate the energy that can be recovered by slowing the vehicle to a speed of 10 km/h.
Belt‐and‐Pulley Systems
1 2‐4 Consider the belt and pulley system in Fig. 2-13. Inertias other than that shown in the figure are negligible. The pulley radius r = 0.09 m and the motor inertia JM = 0.01 kg ⋅ m2. Calculate the torque Tem required to accelerate a load of 1.0 kg from rest to a speed of 1 m/s in a time of 4s. Assume the motor torque to be constant during this interval.
2 2‐5 For the belt and pulley system shown in Fig. 2-13, M = 0.02 kg. For a motor with inertia JM = 40 g ⋅ cm2, determine the pulley radius that minimizes the torque required from the motor for a given load‐speed profile. Ignore damping and the load force fL.
Gears
1 2‐6 In the gear system shown in Fig. 2-14, the gear ratio nL/nM = 3 where n equals the number of teeth in gear. The load and motor inertia are JL = 10 kg ⋅ m2 and JM = 1.2 kg ⋅ m2. Damping and the load‐torque TL can be neglected. For the load‐speed profile shown in Fig. 2-1b, draw the profile of the electromagnetic torque Tem required from the motor as a function of time.
2 2‐7 In the system of Problem 2-6, assume a triangular speed profile of the load with equal acceleration and deceleration rates (starting and ending at zero speed). Assuming a coupling efficiency of 100%, calculate the time needed to rotate the load by an angle of 30o if the magnitude of the electromagnetic torque (positive or negative) from the motor is 500 Nm.
3 2‐8 The vehicle in 2-8Example is powered by motors that have a maximum speed of 5000 rpm. Each motor is coupled to the wheel using a gear mechanism. (a) Calculate the required gear ratio if the vehicle’s maximum speed is 150 km/h, and (b) calculate the torque required from each motor at the maximum speed.
4 2‐9 Consider the system shown in Fig. 2-14. For JM = 40 g ⋅ cm2 and
