Kevin J. McCarthy
As contradictory as it may seem, the key to higher servosystem productivity might come from slowing down. That’s because humans tend to regard most systems as being linear when in many cases they are not. To examine this in detail, consider a simple single-axis, linear-motor positioning stage.
Assume the stage uses a triangular motion profile, accelerating for half the move and decelerating over the remaining half. In addition, the stage has a 100% duty cycle with no pause between moves. Ignoring friction in the guideways for simplification, the resulting system can be characterized by a fairly small set of variables.
Travel, X, and moving mass, m, usually tie directly to the application, although efforts to reduce these factors pay rich dividends in terms of system effectiveness. The move time, t, may depend on economic justifications in terms of units-per-hour throughput, but it’s often tied to marketing decisions as well. Speed limit is usually set by one of three factors; the linear encoder, physical limits of the guideway bearings, or the amplifier supply voltage.
The linear-motor constant, Km deserves a closer look. While key to linear-motor response, Km is often missing from vendor data sheets. This constant, for any given motor, relates the ratio between force output and waste heat generation — an undesirable by-product that limits speed. Most motor advertising emphasizes the linear-motor force constant Kf. This is largely a free variable readily changed by altering coil wire gauge.
The motor constant Km, on the other hand, is defined by the copper volume, packing efficiency, and magnetic field strength, and provides a true indication of linear motor quality. Km is particularly helpful when calculating system response. It varies with motor size and design, and typically ranges from 5 to 25 N/W0.5. Motors with larger numbers offer higher acceleration, speeds, or force, but are physically larger and cost more due to greater magnet and coil volumes.
Consider an application where travel X = 0.1 m, the moving mass m = 10 kg, the move time t = 0.1 sec, Km = 10 N/W0.5, and the system has a speed limit of 3 m/sec and is limited to 100-W continuous power.
For the application, first check the top speed against the stated limit. The top speed in a triangular move is:
V = 2X⁄t
which yields a maximum speed of 2 m/sec. This is comfortably within the system speed limit of 3 m/sec. Next calculate acceleration:
A = 4X⁄t2
This yields an acceleration of 40 m/sec2, or just over 4 g. Both above equations are not the usual ones that spring to mind when making motion calculations, but are timesaving shortcuts for triangular moves compared to splitting the move into two parts.
With the acceleration known, the force required is found from
F = mA
This gives a force of 400 kg-m/sec2, or 400 N. Note that force is continuous in this application during acceleration and deceleration. Next, calculate the power that the linear motor must dissipate:
P = (F⁄Km)2
The waste heat to be dissipated is 1,600 W, which will require a large and expensive motor. Alternately, if the existing 10 N/W0.5 motor (with a continuous power rating of 100 W) and 100 msec move time were to be retained, the system would require a pause between each move of 1.5 sec, for a duty cycle of 1:16 or 6.3%.
This example intentionally uses values that would require a very high-power linear motor. The method works, of course, regardless of the actual numbers, but it is useful to examine how the system scales. The simple, yet key, insight is that acceleration is proportional to the inverse square of the desired move time, and that power is proportional to the square of the force. Since F = mA, that is equivalent to saying that power is proportional to the square of the mass and/or acceleration. Because acceleration is proportional to the inverse square of the move time, and power is proportional to the square of acceleration, combining the two reveals that:
P = 16m2X2⁄Km2t4
In other words, power is proportional to the inverse fourth power of the desired move time. This is a significant observation. Most people tend to view the world as fairly linear, but with fourth-power laws, even subtle changes to a system can have significant effects.
For example, reducing the move time by half — a seemingly reasonable request — will multiply linear-motor power requirements by a factor of 16. Similarly, doubling the time reduces power requirements by a factor of 16. In the example, that would let the original motor work properly.
Because the move time turns out to be a sensitive parameter, a closer review of how it impacts system performance is in order. This requires a look at system power, and throughput expressed in moves/sec. Obviously, a long move time reduces moves/sec. As the move time is shortened, moves/sec rises inversely with time. This can be graphed as a standard k/t curve — a hyperbola. What is not obvious is that this smooth curve runs into a discontinuous inflection point when power equals the power rating of the linear motor. At this point, the moves/sec begins an inverse cubic power crash towards zero. This is clearly shown in the Throughput versus move time graph, where two very different regions of the curve are readily apparent. In the example, moves/sec will be a paltry 0.6 when move time is 100 msec. The rapidly escalating power requirements associated with short move times are also graphed.
The cause of the discontinuity is fairly simple. As the linear-motor coil reaches its continuous power rating, any further move-time reduction requires longer cooling periods to dissipate the excess power. Despite the intuitive conclusion that the delay is offset by faster moves, the fourth power nature of the relationship leads to vastly lower moves/sec.
The most important observation is that for any given system, there is an optimum move time. If a given system falls on the left side of the cusp, increasing the move time will actually increase moves/sec and productivity. A system truly can slow down to speed up. In the example, slowing the move time from 100 to 200 msec increases moves/sec from 0.6 to 5.0 — an 800% improvement.
In many systems, engineers select the move time based on the goal of a certain number of parts/hr, or choose an overall cycle time and then allocate it among the machine operations with little or no attention to proper motor sizing. In some cases, the motor-coil thermal sensors or an amplifier will automatically insert cooling-off pauses, and the only obvious signs of problems are that the system is sluggish and not meeting spec. In other cases, technicians pad in delays on an empirical basis until the linear-motor coils stop burning out. For the simple example, it would be fairly easy to properly size the motor to operate at the desired throughput. In real-world systems with considerably more complex motion sequences, it is a rare designer who prepares a detailed power model of the complete system. Additional constraints can include the cost or the space available for the motor. The concepts expressed in this article also apply to rotary motor systems, although the rotor and leadscrew inertias often predominate over the reflected payload mass.
The inverse fourth-power relationship between power and move time requires a careful consideration of cycle times and motor size when designing motion systems. The square law between moving mass and power requires similar efforts to minimize mass. Casual comments such as “slice 100 msec off the cycle time” need to be evaluated in light of the potentially expensive underlying physics. Finally, the fact that the productivity curve has such a pronounced maximum can lead to more thoughtfully designed systems that operate at peak performance.