# Getting torque-to-inertia right

March 23, 2006
The case of a pulsating-load testing machine shows how to ballpark qualities that are important for deploying electric motors.

Ed Lee
General Manager
Powertec Industrial
Motors Inc.
Rockhill, S.C.

 A 15-hp, E-182 frame brushless-dc motor from Powertec uses neodymium permanent magnets on the shaft and a conventional three-phase stator winding. An internal resolver provides commutation and speed feedback for the drive controller.

Crankshafts in running internal-combustion engines pulsate as well as rotate. This pulsation creates both primary and harmonic torsional stress on crankshaft-driven equipment such as superchargers, pumps, alternators, counterbalancers, and shafts. These components must undergo testing during design and production to check performance and quality. But on-engine component testing is expensive, time consuming, and often impractical. Hydraulic systems and offbalance rigs can simulate engine dynamics. But the equipment is difficult to use, expensive, and need lots of maintenance.

Electric motors would be ideal for such testing. But it is impossible for most electric motors to produce rotational speeds of 500 to 6,000 rpm (or higher), while simultaneously varying those speeds up to ±10% at rates to 100 Hz or more. The reason: the ratio of torque to inertia, which is a measure of motor response.

For example, consider a case where you modulate the rotation of a 1-lb-ft2 load inertia 5% above and below a target speed of 2,000 rpm at 100 Hz. Torque must be applied to repeatedly accelerate and decelerate the load inertia in the allotted time. Assuming uniform acceleration, torque is given by:

T = (I × rpm) / (308 × t)

where T = torque (lb-ft), I = inertia (lb-ft2), and t = time (sec).

In this example, a 25-hp motor can develop the 65-lb-ft continuous torque needed for the job. But the rotor in such an ac induction motor has inertia four times that of the load. In other words, the motor will need 260 lb-ft of torque to overcome its own inertia at the 100-Hz cycle rate. This amount of continuous torque is not available from the motor.

Another way of looking at the problem is to solve the above equation for rpm/sec. A 25-hp motor, based on rated torque and inertia, and with no other inertia attached to the shaft, could accelerate at 20,020 rpm/sec. Of course, the part to be tested also has inertia. It is common engineering practice to match motor and load inertia. Then the total torque is 130 lb-ft. A typical 50-hp NEMA-standard ac-induction vector motor can continuously deliver this amount of torque. But it has a rotor inertia of about 5.5 lb-ft2, significantly higher than the target 1 lb-ft2. In fact, ac-induction motors — regardless of horsepower and torque ratings — cannot respond quickly enough for the application.

Brushless-dc motors, on the other hand, can. The reason: They have significantly higher torque-to-inertia ratios than acinduction or brush-dc motors. A brushless-dc motor sized 50 hp or larger in this case meets specs for both response time and continuous power.

But there is more to system performance than motor torque. The controller must control motor-shaft speed at the desired cycle rate. A rule of thumb says controllers need a velocity-loop bandwidth of at least five times — ideally 10 times — greater than the frequency of motor-shaft speed variation. In this case for 100-Hz variation, the velocity-loop bandwidth should be at least 500 Hz. Similarly, current-loop bandwidth should be five times the velocityloop bandwidth for the controller to work properly, or 2.5 kHz in this application. Moreover, for adequate control of motor current, the pulse-width-modulation rate should be at least five times the current-loop bandwidth.

These values are considered minimums and assume that the load is well characterized and the control algorithm does a loop-ahead calculation on actual output based on an output model (smart controller). Naturally, the question may arise: If smart controllers help brushlessdc motors accurately track control inputs, can they do the same for ac-induction motors? Yes and no. Smart controllers can make ac-induction motors respond faster to control inputs, but they can't adequately overcome high rotor inertia for them to work in this application.

Another measure of drive performance is the rate at which the drive updates analog inputs. Many drives take 1 msec or longer to update analog inputs, excessively long to meet the bandwidth needs in this case. Analog inputs are ±10-Vdc control signals that direct the drive to move a motor in a specified way. The signals may come from a signal generator, a computer program, or from the drive itself if it is programmed internally. Position and velocity feedback typically comes from a digital encoder or resolver.

Analog-to-digital converters should have at least 12-bit resolution to accurately define analog inputs over a broad speed range. A sinusoidal velocity signal processed by an 8-bit converter, in contrast, would pass to the controller as a trapezoid or other distorted shape because the resolution is too coarse.

PUTTING MOTORS TO THE TEST
A torsional test machine inspects automotive superchargers during production at speeds to 6,000 rpm. Torsional-vibration tests take place at 2,000 rpm and vary speed ±5% at 60 Hz. Total inertia is about to 1 lb-ft2. A brushlessdc motor rated at 60 hp at 6,000 rpm can do the job.

Another rig tests IC-engine counterbalancers. Such devices install on the crankshaft to counminimums teract primary and secondary torsional vibration from piston forces. This application varies rotation velocity ±5% about 1,200 rpm at 100 Hz. This application is impossible with motors having a torque-to-inertia ratio less than 60 lb-ft / lb-ft2.

Considering that the load itself has some inertia, motor inertia must be as small as possible, to a point. It is generally agreed for stability reasons that motor inertia should not be less than 20 to 25% of load inertia. Systems that match motor and load inertia consume the least amount of power and best track input signals (highest system bandwidth).

 Plot of command and actual motorshaft velocity. The motor accurately follows an analog velocity command variation input. The controller accepts an analog input voltage and produces a smooth sinusoidal output with effectively no phase delay.

 TYPICAL PERFORMANCE OF AC INDUCTION AND BRUSHLESS-DC MOTORS Horsepower Rated torque (lb-ft) Frame size Inertia (lb-ft2) Frame size Inertia (lb-ft2) —— ac induction —— —— brushless dc —— 10 30 215T 1.1 E182T 0.3 15 45 256T 1.8 ES184T 0.4 20 60 256T 2.3 E184T 0.5 25 75 284T 4.0 E213T 0.8 30 90 286T 4.7 E213T 0.8 40 120 324T 7.8 E215T 0.9 50 150 326T 9.7 E218T 1.1 60 180 364T 12.2 E254T 2.4 75 225 365T 15.3 E256T 2.9 100 300 405T 27.0 ES25T 3.8

MAKE CONTACT Powertec Industrial Motors Inc., www.powertecmotors.com

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