The Natural Period in .NET

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The Natural Period
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It has also been observed that the period of oscillation which a particular loop will exhibit is characteristic of that loop. The loop resonates at that period. Furthermore, any disturbance not periodic, applied to the loop but containing components near the natural period, will excite oscillations of the natural period. A pendulum is a good example of a feedback loop. The controlled variable is the angular position of the mass, and the set point is the vertical position. The mass of the pendulum, acted upon by gravity, is the manipulated variable, which tries to restore the angle to zero. Its natural period in seconds is 1 L $6 7 o=- 27r 0 9 where I, = length, ft g = acceleration of gravity, ft/sec2 A pendulum disturbed from rest by an impulse will proceed to oscillate at its own period. Impulse, step, and random disturbances contain a wide spectrum of periodic waves. The resonant system, however, responds only to the component of its own natural period, rejecting the rest. For this reason, we are interested in the response of the loop to a wave of the natural period and are generally unconcerned about the rest. The natural period of oscillation will be designated 70 and will be recognized hereafter as a property peculiar to each control loop. The natural period of any loop depends on the combination of all dynamic elements within it, including the controller. Since the amount of phase lag of most dynamic elements varies with the period of the wave passing through them, there is one particular period at which the total phase lag will equal 180 . This is the period at which the loop naturally resonates. The natural period is a dependent variable. We can make use of its relation to the process dynamics in two ways: 1. If the characteristics of the elements in the process are known, the natural period under closed-loop control can be predicted.
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1 Ud erstanding Feedback Control n
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2 . If a process whose elements are largely unknown is under closed-loop control, the characteristics of these elements can be inferred by observing the natural period.
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The gain of an element is defined as the ratio of the change in its output to the change in its input. If the controller gain were zero, it would not contribute to oscillation. But if the controller gain were sufficient to produce a second disturbance equal to the first, the loop would oscillate uniformly. Uniform oscillation requires that a wave travel completely through the loop, returning to its starting point with its original amplitude. For such a condition to exist, the gain product of all the elements in the loop must equal unity. If the gain product is less than unity, oscillations are damped. To summarize, a loop will oscillate uniformly: 1. At a period at which the phase lags of all the elements in the loop total 180 2. When the gain product of all the elements at that period equals 1.0 The conditions for uniform oscillation will serve as a convenient reference on which to base rules for controller adjustment.
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As the name implies, dead time is the property of a physical system by which the response to an applied force is delayed in its effect. It is the interval after the application of a force during which no T.esponse is observabIe. This characteristic does not depend on the nature of the applied force; it always appears the same. Its dimension is simply that of time. Dead time occurs in the transportation of mass or energy along a particular path. The length of the path and the velocity of motion consti-
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