Time domain substitute

We will Laplace transfonn Cao<,), substitute into the system transfer function, solve for Ca(,)> and invert back into the time domain to find... 

The ordinary diflcrentiaJ equations could be solved in the time domain with a sinusoidal forcing function, but it is easier to go into the Laplace domain first and then substitute s = iw. 

Equation (18.4) expresses the sequence of impulses that comes out of an impulse sampler in the time domain. Equation (18.5) gives the sequence in the Laplace domain. Substituting io) for s gives the impulse sequence in the frequency domain. 

Figure 4. Time domain trace resulting from the mechanical disbonding of a chlorinated rubber substitute.

It is sometimes more conventional to incorporate the local temperature fluctuations as well into A, in particular when one wishes to relate the resulting theory with hydrodynamics. However, the essential points of the theory would not be altered with a simplified choice of the dynamical variables.) Substituting Eq. (5.22) into Eq. (5.11) yields the following GLE for F k,t) in the time domain [18, 19, 20] ... 

The primary photoexcitation dynamics in PDPA solutions and films in the fs to ps time domain using transient PM spectroscopy were extensively studied [182]. The PDPA polymer used was a disubstituted biphenyl derivative of frans-polyacetylene, where one of the hydrogen-substituted phenyl groups was attached to a butyl group, which is referred to as PDPA-mBu (Figure 22.25 inset) [181]. The polymer films were cast on sapphire substrates from a toluene solution the same solution was used for measuring the photoexcitation dynamics in a PDPA-mBu solution. 

The generalization of Eq. (5.3) is given as Eq. (5.14). The variables / and (O denote spectral variables such as frequencies, that may be obtained from any kind of spectroscopy. They are only related by the common time domain t, which can also be substituted by another perturbation dimension, such as a series of samples. 

Now, to find the current/voltage relationship in the time domain, we only need to substitute V(5) for the particular excitation of interest and carry out the inverse Laplace transform for the entire equation. However, first we need to know how the particular excitation we want to apply looks in the Laplace domain. Here is an example of how to do it for a voltage step. In the time domain it can be expressed as v(0 = Heaviside(t) Vi where the function Heaviside(/) takes a value 1 if i > 0 and 0 otherwise. Vi is the value of voltage applied during the step. The Laplace transform of this function, V s) =. [Heaviside(i) Vj] = VJs. To find the Laplace current of the above mentioned network to pulse excitation, we substitute this V(x) into Eq. (4) and perform inverse Laplace transformation of the resulting equation. We obtain, ... 

Taking note of the implications of equations (7) and (8), the frequency-domain filter can be written down directly from the time-domain version (equations (4) and (5)) simply by substituting the frequency interval n in place of the time interval T. 

Back transformation from the transfer function to a differential equation in the time-domain, can be achieved by substitution of 5 -x(5 ) = dx(t)/dt. The variable s is independent of the time and indicates more or less the rate of change. This can be explained by the final-value theorem and the initial value theorem apphed to a variation in the input variable of a system. 

If Equation 5.63 is substituted into Equation 5.60, the electric-field integral equation in the time domain for a thin-wire perfect conductor is obtained as... 

The solution of the wave propagation problem in viscoelastic media in the time domain which follows from the direct substitution of Eq. 36 in Eq. 1 is, however, inconvenient because that would entail the computation of a convolution term at every time step. Such an approach would require the storage of the complete strain... 

Here the output voltage and input displacement Xi are written in uppercase to emphasize that this transfer function operates in the frequency domain. It is a property of the Laplace transform that multiplication by the (complex) frequency s in the frequency domain is equivalent to differentiation in the time domain, so sX, is the input velocity. Another property of the Laplace transform is that to evaluate the (complex) transfer function at a given frequency f in Hz or angular frequency co = 2nf in rad/s, one makes the substitution s = jm. 

Most of the photophysical data for azulene and its simple derivatives have come from measurements of the quantum yields of Sj - Sq fluorescence, ( )s2, and time-domain measurements of the lifetimes of the fluorescent Sj state, Xjj, in fluid solution. The first order rate constants for the parallel radiative and nonradiative processes that depopulate Sj are then determined from k = ( )s2/Xs2 and Ek, = (1 - ( )s2)/ s2- Whereas Sj - Si internal conversion is clearly the major, and perhaps the exclusive, pathway for Sj s nonradiative decay in azulene itself, other nonradiative processes can be expected to occur in some substituted azulenes. 

---