Unit step function

The unit step function s(x) is defined as a step from 0 to /. The function s(x) is shown in fig. 1 (centre) together with an example of a step response function h(x). 

Unit impulse fucnction Unit step function Example for an input signal 5(x) f s(x)f u( ),... 

Table 7.2 shows the discrete response x ikT) to a unit step function and is compared with the continuous response (equation 3.29) where... 

Figure C.l The Heaviside unit step function H x), defined as the limit as e — 0 of...

We first define the unit step function (also called the Heaviside function in mathematics) and its Laplace transform 1... 

The Laplace transform of the unit step function (Fig. 2.3) is derived as follows ... 

In mathematics, the unit rectangular function is defined with a height of 1/2T and a width of 2T from -T to T. We simply begin at t = 0 in control problems. Furthermore, the impulse function is the time derivative of the unit step function. 

Example 2.10 What is the time domain response C (t) in Eq. (2-27) if the change in inlet concentration is (a) a unit step function, and (b) an impulse function ... 

For PF, the F function requires another type of special mathematical representation. For this, however, consider a sudden change in a property of the fluid flowing that is maintained (and not pulsed) (e.g., a sudden change from pure water to a salt solution). If the change occurs at the inlet at t = 0, it is not observed at the outlet until t = t. For the exit stream, F(t) = 0 from ( = 0 to t = t, since the fraction of the exit stream of age less than ( is 0 for t < f in other words, the exit stream is pure water. For t > t, F(t) = 1, since all the exit stream (composed of the salt solution) is of age less than t. This behavior is represented by the unit step function S(t - b) (sometimes called the Heaviside unit function), and is illustrated in Figure 13.7, in which the arbitrary constant b = t. With this change, the unit step function is... 

A forcing function, whose transform is a constant K is applied to an under-damped second-order system having a time constant of 0.5 min and a damping coefficient of 0.5. Show that the decay ratio for the resulting response is the same as that due to the application of a unit step function to the same system. 

Example 6.11. The overdamped system of Example 6.8 is forced with a unit step function. 

Example 6il2. A general underdamped second-order system is forced by a unit step function ... 

Now the unit step function can be expressed as a limit of the first-order exponential step response as the time constant goes to zero. 

The 1/s is an operator or a transfer function showing what operation is performed on the input signal. This is a completely different idea than the simple Laplace transformation of a function. Remember, the Laplace transform of the unit step function was also equal to l/s. But this is the Laplace transformation of a function. The 1/s operator discussed above is a transfer function, not a function. 

Inversion of the Laplace transformation gives the time function. If we have a transfer function the unit step function is C [G(,y s] and the impulse response is C" [G(,)]. 

Velocity as a function of radial position Mean velocity in a packed bed based on empty tube Unit step function Volumetric flow rate Volume of vessel Volume of backmix flow region... 

To formulate the problem in a convenient way, we consider that during a scan, the contour of the sample surface has a unit jump at r = 0. In other words, the height is assumed to be a unit step function. 

We do not lose generality by considering such a unit step function. Because the differential equation is linear, by making superposition of the step function, the response from any surface contour can be treated. The Laplace transform of a step function is... 

The Dirac delta function 5(f) localizes the integration onto the dividing surface f= 0. The gradient of the dividing sluface (Vf) is in the full phase space,pis the generalized velocity vector in phase space with components q, Pq, xj, p, ) = 1,.. ., N, and 0(y) is the unit step function which restricts the flux to be in one direction only. The term V f p is proportional to the velocity perpendicular to the dividing smface. The munerator is the imidirectional flux and the denominator is the partition fimction of reactants. 

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