Laplace transforms unit impulse

The Laplace transform of the impulse function is obtained easily by taking the limit of the unit rectangular function transform (2-20) with the use of L Hospital s rule ... 

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(,)]. 

Defining in this way permits us to use transfer functions in the z domain [Eq. (18.57)] just as we use transfer functions in the Laplace domain. G,, is the z transform of the impulse-sampled response of the process to a unit impulse function <5( . In z-transforming functions, we used the notation =... 

The term in parentheses is the Laplace transformation of the impulse-sampled response of the total combined process to a unit impulse input. We will call this (GiG,)S, in the Laplace domain and (Gi G2)(i) in the z domain. 

The exit concentration Cm(z — 1, t) for the case of a unit impulse (Delta function) input (f(z) — 0, git) — S(i)) is known as the dispersion (or RTD) curve. For the hyperbolic model, this can be found either by Laplace transformation or from the general solution of the model [see Balakotaiah and Chang (2003), for a general analytical solution of Eqs. (57)—(59)]. It is easily seen that the Laplace transform of the dispersion curve is given by... 

Consider an impulse function. The impulse function is also known as a Dirac delta function and is represented by S(t). The function has a magnitude oo and an area equal to unity at time t = 0. The Laplace transform of an impulse function is obtained by taking the limit of a pulse function of unit area ast 0. Thus, the area of pulse function HT = 1. The Laplace transform is given by... 

The perturbed unit impulse response UIR is determined by recomposition of the disposition kinetics from the distribution function and the perturbed clearance via Laplace transformation. 

The UIR function is also commonly denoted the characteristic function. This is an appropriate notation since it is a function that is characteristic for the given input-response system. In engineering, the Laplace transform of the unit impulse response is commonly called the transfer function. The above equations for x t) and c(0 can be written ... 

Z(w) = 1/wC, and in the s-domain Z(s) = 1/sC. The Laplace transforms of some very important excitation waveforms are very simple for example, for a unit impulse it is 1, a unit step function 1/s, a ramp 1/s, etc. That is why the excitation with, for example, a unit impulse is of special interest examining the response of a system. In the extended immittance definition, calculations with some nonsinusoidal waveforms become very simple. Even so, Laplace transforms are beyond the scope of this book. 

Here c is the local tracer concentration in environmental i,Ui,and are the velocity vector and diffusivity in environment i, and t is the residence time in environment i. Proper boundary conditions require no accumulation of tracer at the boundaries and continuity of tracer flux. Closedness of the system on the boundaries with the inlet and exit environment is required also, i.e. the net input and output of tracer occurs by flow only. To obtain directly from the model the joint p.d.f. a normalized unit impulse input is required. First of all it is readily apparent that if we take the Laplace transform of the above equation we get ... 

Impulse Function. A limiting case of the unit rectangular pulse is the impulse or Dirac delta function, which has the symbol 8(r). This function is obtained when ty while keeping the area under the pulse equal to unity. A pulse of infinite height and infinitesimal width results. Mathematically, this can be accomplished by substituting h = lltyy into (3-22) the Laplace transform of b(i) is... 

Impulse Input. The unit impulse function discussed in Chapter 3 has the simplest Laplace transform, U s) = 1 (Eq. 3-24). However, true impulse functions are not encountered in normal plant operations. To obtain an impulse input, it is necessary to inject a finite amount of energy or material into a process in an infinitesimal length of time, which is not possible. However, this type of input can be approximated through the injection of a concentrated dye or other tracer into the process (see Example 3.7). 

Next, we derive the Laplace transform of Eq. 17-20, y (5 ). The value of y(kAt) is considered to be a constant in each term of the summation and thus is invariant when transformed. Since 5S[S( )] = 1, it follows that the Laplace transform of a delayed unit impulse is 5 [b(t - kAt)] = Thus, the Laplace transform of... 

A function obtained by the inverse Laplace transform of the transfer function of a dynamic system is equivalent to the impulse response of the system. Therefore, the shear stress variation at the adhesive edge Ta imp(0, f) by a unit impulse stress wave can be written as the following equation. 

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