Equation system frequency response

To model the system frequency response, we rewrite the PDEs, represented by equation (8.43), by replacing u(x,t) with u<,(x)+u (x,f), the steady state solution plus a small time varying perturbation u. The steady state and the perturbation were set by the input current density j t)= Jo + / (0- Note we could have perturbed the other inputs and Xq, but this is not experimentally practical. We substitute u +u into equation (8.43) and express the c matrix as C1U+C2. This is to cover the nonlinear situation in the migration terms containing y j d /dx and y dX/dx in equation (8.45). 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

In the limit of small pressure perturbations, any kinetic equation modeling the response of a catalyst surface can be reduced to first order. Following Yasuda s derivation C, the system can be described by a set of functions which describe the dependence of pressure, coverage amplitude, and phase on T, P, and frequency. After a mass balance, the equations can be separated Into real and Imaginary terms to yield a real response function, RRF, and an Imaginary response function, IRF ... 

As you will see, several different approaches are used in this book to analyze the dynamics of systems. Direct solution of the differential equations to give functions of time is a time domain teehnique. Use of Laplace transforms to characterize the dynamics of systems is a Laplace domain technique. Frequency response methods provide another approaeh to the problem. 

There are a number of ways to obtain the frequency response of a process. Experimental methods, discussed in Chap. 14, are used when a mathematical model of tbe system is not available. If equations can be developed that adequately describe the system, the frequency response can be obtained directly from the system transfer function. 

The Nyquist stability criterion is, on the surface, quite remarkable. We are able to deduce something about the stability of the closedloop system by making a frequency response plot of the openloop system And the encirclement of the mystical, magical (— 1, 0) point somehow tells us that the system is closedloop unstable. This all looks like blue smoke and mirrors However, as we will prove below, it all goes back to finding out if there are any roots of the closedloop characteristic equation in the RHP. 

All the Nyquist, Bode, and Nichols plots discussed in previous sections have been for openloop system transfer functions B(j ). Frequency-response plots can be made for any type of system, openloop or closedloop. The two closedloop transfer functions that we derived in Chap. 10 show how the output is affected in a closedloop system by a setpoint input and by a load. Equation (13.28) gives the closedloop servo transfer function. Equation (13.29) gives the closedloop load transfer function. 

The frequency response of a system may be more easily determined by effecting the substitution s = iw in the appropriate transfer function [where i = y/(—1)] (i.e. mapping the function from the s domain onto the frequency domain(17)). Hence substituting into equation 7.19 ... 

It can be shown(18) that this method may be applied to any system described by a linear differential equation or to a distance-velocity lag in order to obtain the relevant frequency response characteristics. 

Systems in series. The usefulness of the logarithmic plot becomes apparent when it is desired to determine the frequency response of systems in series. The resultant amplitude ratio and phase shift may be obtained using equations 7.104 and 7.105, i.e. ... 

Herein lies an opportunity for computing excitation spectra (and the actual CD intensity) from TDDFT linear response Once a response equation for /i(ffl) (or 4>(a>)]) has been derived, circular dichroism can be computed from an equation system that determines the poles of [> on the frequency axis, just like regular electronic absorption spectra are related to the poles of the electronic polarizability a [27]. Details are provided in Sect. 2.3. We call this the linear response route to calculating excitation spectra, in contrast to solving (approximations of) the Schrodinger equation for excited state and explicitly calculating excited state... 

Consider, for example, a test sample of material with a well-defined geometry as shown in Fig. 2. Reversible electrodes are attached to opposite planar faces of the test article, and a sinusoidal electrical potential (V ) is applied via a waveform generator. The current response is monitored with a frequency response analyzer (FRA), which converts the signal to the frequency domain. The amplitude (A) of the input wave is adjusted to the range in which the system responds linearly, about 10 mV. Thus, the perturbation can be described by the following equation ... 

By the start of World War II, a new approach to control system synthesis was being developed from Nyquist s theoretical treatment (Nl) of feedback amplifiers in 1932. This approach utilized the response of components and systems to steady-state sinusoidal excitation or frequency response as it is more usually called. The frequency response approach provides an important basis for present-day methods of handling control problems by affording a simply manipulable characterization which avoids the need for obtaining the complete solutions of system equations. 

With nonlinear systems, however, all simplicity disappears. No general methods of solving even the simplest, nonlinear differential equations are known. Frequency response characterization is useless since sinusoidal forcing will not produce sinusoidal response. The only recourse other than arbitrary linearization of the equations is to utilize... 

The use of long leads between the potential source and the measuring instrument can result in an effective change of the output capacitance of the measuring instrument, thus altering its frequency response. Typically the capacitance of a twin core cable is of the order of 100 pF/m. The effect on the frequency response can be calculated using equations 3 or 4. The remedy to this problem is to keep the cables as short as possible or, where long cable systems are unavoidable, a driver amplifier at the source may be required. 

In evaluating control systems for microrotorcrafts, Mettler [8] found that frequency domain identification served as a better method than time domain identification. This is because in frequency domain identification, the output measurement noise does not affect the results, it is possible to focus on a precise frequency range (which minimizes the disparity between the modes of motion), and frequency responses can completely describe the system s linear dynamics. Mettler also determined that the rigid body equations of motion needed to be expanded through use of the hybrid formulation to generate a more accurate control system. This method models the rotor motion using a tip-path plane model and expresses the rotor forces and moments in terms of the rotor states. The rotor and fuselage motions are then dynamically coupled. 

These equations indicate that any bandpass filter systems may be described and analyzed by using an equivalent low-pass filter as shown in Fig. 12.3(c). A typical equivalent low-pass frequency response characteristic is shown in Fig. 12.3(d). Since equations for equivalent low-pass filters are usually much less complicated than those for bandpass filters, the equivalent low-pass filter system model is very useful. This is the basis for computer programs that simulate bandpass communication systems by using equivalent low-pass models. 

The materials and structures associated with primary sensors contain dissipative, storage and inertial elements. These translate into the time derivatives appearing in the differential equation that models the sensor system. Hence another major defect is represented by the time (or frequency) response. The means to neutralise this imperfection involves filtering, which may be thought of in terms of pole-zero cancelation. If the device has a frequency response H s) then a cascaded filter of response G s) = 1/H s) will compensate for the non-ideal time response. The realisation of such a filter in analogue form presents a major obstacle that is greatly diminished in the digital case. 

Based on the standard equation of motion, Eq. (A6), the complex frequency response function and the impulse response function hy Ct) for the relative displacement Y Ct) of the coupled system are... 

The issue of downstream boundary conditions becomes a bit more complex when heat transfer and solidification are included, since the solidification point will move dynamically when the system is perturbed. The transform methodology requires that the frequency response equations be solved on a fixed spatial domain. We can retain this structure as long as we retain the simplified condition that solidification occurs when the temperature reaches a fixed value in that case the dynamics can be linearized about the solidification point, and the downstream boundary condition can be written in terms of a linear combination of the velocity and temperature perturbations at the steady-state solidification point. (This issue does not arise with the approach used by McHugh and co-workers, in which crystallization kinetics are included and the fixed domain is the entire spinline.)... 

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