Laplace frequency variables

The stability of the reactor may be investigated by means of equation (5) if the kernel A( ) is known. However, it must be remembered that stability against small oscillations does not guarantee stability against finite oscillations as we shall see later. Thus it is necessary also to investigate the nonlinear problem in order to insure the stability of the reactor. In treating the theory of small oscillations it is most convenient to work in frequency space, or in terms of the Laplace transform variable s. The feedback term of the first equation (5) then takes the form ... 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

Consider first the series junction of N waveguides containing transverse force and velocity waves. At a series junction, there is a common velocity while the forces sum. For definiteness, we may think of A ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance Rj (s), as depicted in Fig. 10.11 for TV= 4. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., V(s) = C v for velocity waves and F(s) = C / for force waves, where jC denotes the Laplace transform. In the discrete-time case, we use the z transform instead, but otherwise the story is identical. 

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

The Laplace variable 5 is a frequency parameter that characterizes the time scale. As long as the times ofinterest are long compared to ft>o ( 10 sec), the lower limit of integration in Eq. (AlO) can be taken to be zero (in the a < 1 limit). In this limit the integral can evaluated analytically and Eq. (A4) becomes... 

The complex variable z (Im z < 0) is homogenetic to a frequency. The resolvent l/(z — L) is the Fourier-Laplace transform of the evolution operator (see Appendix A). Expression (93) shows that the dynamics is reduced to the determination of the matrix element of the resolvent between two observables. Therefore only a reduced dynamics has to be investigated. For that purpose we shall define more precisely the observables and the operators of interest. The theory is formulated in the framework of the Liouville space of the operators and based on hierarchies of effective Liouvillians which are especially convenient to study reduced dynamics at various macroscopic and microscopic timescales (see Appendix B). 

We make now, similarly as is common with the different integral transforms, a correspondence table between the stochastic variable and the associated characteristic function. Note, there are several integral transforms. The most well-known integral transformation might be the Fourier transform. Further, we emphasize the Laplace transform, the Mellin transform, and the Hilbert transform. These transformations are useful for the solution of various differential equations, in communications technology, all ranges of the frequency analysis, also for optical problems and much other more. We designate the stochastic variable with X. The associated characteristic function should be... 

Discrete frequency Discrete Fourier transform (DFT) Fourier analysis, periodic waveform Continuous frequency Discrete Fourier transform (DTFT) Fourier transform Continuous variable z-transform Laplace transform... 

A more general mathematical treatment is possible by introducing the complex frequency s = a + jw, allowing sine waves of variable amplitude, pulse waveforms etc. Such Laplace analysis is outside the scope of this book, however. 

Usually, the Laplace transform is written with a real or complex variable s that we have replaced by the product of the imaginary times the angular frequency, as frequently done.) Some characteristic transformed functions are worth citing. 

Up to this point we have described methods in which impedance is measured in terms of a transfer function of the form given by Eq. (56). For frequency domain methods, the transfer function is determined as the ratio of frequency domain voltage and current, and for time domain methods as the ratio of the Fourier or Laplace transforms of the time-dependent variables. We will now describe methods by which the transfer function can be determined from the power spectra of the excitation and response. 

Establish directly by solving Eq. (5.2.16) via the method of Laplace transforms for the case of constant aggregation frequency, given by a x, x ) = the self-similar solution il/ rj) = e. (Hint Recognize the convolution on the right-hand side of (5.2.16). Letting = ij/ where ij/ is the derivative of the Laplace transform ij o ij/ respect to the transform variable s, obtain and solve a (separable) differential equation for the derivative of 0 with respect to ij/). 

In the Laplace or z-transform domain, these input-output models usually occur in what is known as the transform domain transfer function form in the frequency domain, they occur in the frequency-response (or complex variable) form in the time domain, they occur in the impulse-response (or convolution) form. 

Now, we Laplace transform the primed variables and their time derivatives from the time domain t to the frequency domain co... 

By using frequency sweep, the parameters Ri, R2 and C can be found. In sinusoidal steady-state, the Laplace variable s = ju) possibilities to rewrite (1) as... 

Frequency response analysis is another classical tool that has been used in the analysis and design of process control systems. The Laplace variable s is replaced by jfrequency response is then plotted using an Argand diagram approach. 

The linearized and Laplace-transformed equations of the models described above are used to evaluate the various system transfer functions as functions of the Laplace variables s = cr + jco, where a is the real part and co is the imaginary part of the complex variable s. a refers to the damping constant (or damped exponential frequency) and co refers to the resonant oscillation frequency of the system. 

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