Time-domain response

Time domain response of first-order systems 3.5.1 Standard form... 

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 ... 

Time integral performance criteria (ISE, IAE, ITAE) Apply design relations derived from minimization of an error integral of the theoretical time-domain response. 

Since the model is stable, all the roots of P(s), whether they be real or complex, have negative real parts and their corresponding time domain terms will decay away exponentially. Thus if we are only interested in the time domain response at sufficiently large times, we only need to consider the partial fraction expansion of the two purely sinusoidal terms associated with the input ... 

We can design a controller by specifying an upper limit on the value of Mpco. The smaller the system damping ratio C the larger the value of Mpco is, and the more overshoot, or underdamping we expect in the time domain response. Needless to say, excessive resonance is undesirable. 

Where do we go from here We may stay with the design or we may increase the margins We also can use MATLAB to simulate the closed-loop time domain response and from the... 

MATLAB should return Kcu = 40.2 (32.1 dB). Thus following Eq. (8-24), we need to use Kc = 40.2/2 = 20.1 to meet the gain margin specification of 2. You can double check the result yourself with kc=2 o. l, taui=l. 5 8. If so, you should find that the phase margin is now 23°—a bit low but we probably can accept that. After this, we may proceed to the time domain response calculations. 

Note In the text, we emphasize the importance of relating pole positions of a transfer function to the actual time-domain response. We should get into the habit of finding what the poles are. The time response plots are teaching tools that reaffirm our confidence in doing analysis in the Laplace-domain. So, we should find the roots of the denominator. We can also use the damp () function to find the damping ratio and natural frequency. 

Fourier transform refers to NMR experiments in which the time domain response of the spin system is transformed into a frequency spectrum. 

Time-domain response of feedback amplifiers has been regularly correlated with frequency-domain behavior, and vice versa. Examples have usually been restricted to those situations in which only the amplifier contributes phase shift (single pole) or where a second source of phase was included (two-pole), such as from nonideal amplifier design or from the effects of stray capacitance at the input terminal. The system of interest in electrochemistry is more complicated than a two-pole system because there is also a decreasing phase shift caused by... 

Time-domain response to a step change is given in Figure 7.21 for the same system. Again the first three curves are identical with Ee/V, curves of Figure 7.12 and the fourth results when F = 1. That the system is teetering on the edge of instability is very evident from curve D. 

The response of the controlled variable to different types of perturbation (forcing function) in set point or load can be determined by inverting the appropriate transform (e.g. equation 7.112). This is possible only for simple loops containing low order systems. More complex control systems involving higher order elements require a suitable numerical analysis in order to obtain the time domain response. 

Figure 3. The acquired NMR signal (a), Free Induction Decay (FID) in the time domain response.Two transient signals each 90° out of phase comprise the real and imaginary part of the FID (b), its frequency domain response using Fburier transformation.

FID). FID is a time domain response of all nuclei irradiated by r.f. excitation pulse and can be analyzed by Fourier transformation into the frequency domain. 

Multiple-point fluorescent deteclion has been proposed to enhance detection sensitivity. This method is based on the use of a detector function, such as the Shah function. The time-domain signals were first detected, and they were converted into a frequency-domain plot by Fourier transformation. Therefore, this technique was dubbed Shah convolution Fourier transform detection (SCOFT). As a comparison, the single-detection point time-domain response is commonly known as the electropherogram [698,699,701]. 

For the time domain responses of the CPE, the current density difference (A/)/scan rate (v) relation is expressed by the following power-law during the potential scanning199... 

To provide an example of the two-dimensional response from a system containing well-defined intramolecular vibrations, we will use simulations based on the polarized one-dimensional Raman spectrum of CCI4. Due to the continuous distribution of frequencies in the intermolecular region of the spectrum, there was no obvious advantage to presenting the simulated responses of the previous section in the frequency domain. However, for well-defined intramolecular vibrations the frequency domain tends to provide a clearer presentation of the responses. Therefore, in this section we will present the simulations as Fourier transformations of the time domain responses. Figure 4 shows the Fourier transformed one-dimensional Raman spectrum of CCI4. The spectrum contains three intramolecular vibrational modes — v2 at 218 cm, v4 at 314 cm, and vi at 460 cm 1 — and a broad contribution from intermolecular motions peaked around 40 cm-1. We have simulated these modes with three underdamped and one overdamped Brownian oscillators, and the simulation is shown over the data in Fig. 4. 

Once the diffusive reorientation contribution has been subtracted from the deconvolved time-domain response, a final Fourier transform yields the intermolecular spectrum (often referred to as the reduced spectral density). 

In a theoretical treatment of time domain responses, van Gemert and de Graan used a reverse procedure, calculating P(<) from a known c(/(u) behaviour. This treatment is idealized because it was necessary to assume a perfect step pulse with zero rise-time. Using Laplace transform... 

Figure 8 Linear response. (A) Analogous models. (B) Time-domain response. (C) Frequency-domain response.

To summarize, all of the information of the system is available from either means of exciting the resonance, driving it and sweeping the frequency, or hitting it with an impulse for its time response. The first experiment is performed in the frequency domain and the second in the time domain. The mathematical transformation of one representation into the other is the Fourier transform. The time domain response and the frequency domain response are called Fourier transform... 

We will now investigate the relationship between frequency domain and time domain responses. In most cases considered in this section a transient electromagnetic field is excited by a step function current in the source. Moreover the theory of the transient induction logging described in this monograph will be developed for this type of excitation. For this reason the relationship between frequency response and transient response corresponding to this single type of excitation will be our principal concern. The information we need is obtained through use of the Fourier transform which takes the well... 

Figure 12.5 Typical time-domain response of stable and unstable systems...

The time-domain response of the linear system to this sinusoidal input can be calculated analytically, and the fraction of time that the output variable is outside a given specification range can be easily computed. Then the profitability is determined using this fraction of on-specification product and combining the conventional capital, energy, and raw material costs. 

The key slice/projection theorem was first formulated in a radio astronomy context by Bracewell [5] and later exploited in NMR by Nagayama et al. [6] and Bodenhausen and Ernst [7, 8]. Consider the case of a typical plane 5(Fi 2) from a three-dimensional NMR spectrum S Fi,F2,F ). In order to obtain a projection at some angle a, the theorem postulates that the time domain response should be sampled along a slice through the origin at this same angle a. This requires that the evolution parameters C and t2 be varied jointly [7-13] ... 

Transient time-domain response to impulse excitation... 

Historically, most of the spectral responses discussed in the succeeding chapters first came into general use in the form of a steady-state response (usually absorption-mode rather than disper-sion-mode) to a "continuous-wave" oscillating driving force. More recently, the same information has come to be extracted from the time-domain response of the same system to a sudden impulse. 

In the previous section, we established a correspondence between the transient time-domain response (exponentially damped cosine wave) to a sudden "impulse" excitation and the steady-state frequency-domain response (Lorentzian absorption and dispersion spectra) to a continuous excitation. The Fourier transform may be thought of as the mathematical recipe for going from the time-domain to the frequency-domain. In this section, we shall introduce the mathematical forms of the transforms, along with pictorial examples of several of the most important signal shapes. 

Figure 9. Apodizations of chi experimental FT-NMR signal. (A) Fourier transform of the original unweighted free induction decay (F.I.D.) time-domain response to a 90 -pulse excitation. (B) Signal-to-noise enhancement F.I.D. weighted by the factor, exp(- r>LB-t), with LB = 3.0 Hz, before F.T.

Deconvolution of response to frequency-sweep excitation, (a) Cosine Fourier transform of linearly increasing frequency sweep time-domain waveform, (b) Magnitude spectrum of excitation, (c) Cosine Fourier transform of time-domain response to excitation, (d) Magnitude spectrum of response, (e) = (c)/(a). (f) Magnitude spectrum... 

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