Sine transfer function

Fig. 2 Phase-plane output for the transfer function with a sine wave disturbance at a frequency of F = 0.1.

J. The frequency-response data given below were obtained from direct sine-wave tests of a chemical plant. Fit an approximate transfer function to these data. 

The first term on the right-hand side of eqn. (11) decays away and, after a time approximately equal to 5t, the second term alone will remain. Note that this is a sine wave of the same frequency as the forcing function, but that its amplitude is reduced and its phase is shifted. This second term is called the frequency response of the system such responses are often characterised by observing how the amplitude ratio and phase lag between the input and output sine waves vary as a function of the input frequency, k. To recover the system RTD from frequency response data is more complex tnan with step or impulse tests, but nonetheless is possible. Gibilaro et al. [22] have described a short-cut route which enables low-order system moments to be determined from frequency response tests, these in turn approximately defining the system transfer function G(s) [see eqn. (A.5), Appendix 1]. From G(s), the RTD can be determined as in eqn. (8). 

Fig. 26 Fourier transform spectrum of v2 of ammonia. Trace (a) is a section of the infrared absorption spectrum of ammonia recorded on a Digilab Fourier transform spectrometer at a nominal resolution of 0.125 cm-1. In this section of the spectrum near 848 cm-1 the sidelobes of the sine response function partially cancel, but the spectrum exhibits negative absorption and some sidelobes. Trace (b) is the same section of the ammonia spectrum using triangular apodiza-tion to produce a sine-squared transfer function. Trace (c) is the deconvolution of the sine-squared data using a Jansson-type weight constraint.

Figure 7.2 (A) Voltage-transfer function and phase lag of equivalent cell with RsCdl = 1 x 10"4 s and RuCd, = 1 x 10 5 s. VD is amplitude of sine wave of frequency f. (B) Transient response of same equivalent cell. V0 is magnitude of potential step applied at t = 0.

Impedance transfer functions may be determined through use of an input signal containing more than a single frequency. Such signals may be a tailored multi-sine... 

Although the transfer function gives a complete dynamic description of process elements, some interpretation is necessary. The response of the element to any kind of forcing function can be determined from the transfer function, but only a very few kinds of forcing are of any importance in control problems. These important forcing functions are (a) step (b) pulse (c) ramp (d) steady state sine wave and (e) random. 

Now let us prove that this simple substitution s = io) really works. Let G(s) be the transfer function of any arbitrary Nth-order system. The only restriction we place on the system is that it is stable. If it were unstable and we forced it with a sine wave input, the output would go off to infinity. So we cannot experimentally get the frequency response of an unstable system. This does not mean that we cannot use frequency-domain methods for openloop-unstable systems. We return to this subject in Chapter 11. 

The frequency transfer function SHM method resembles the vibration SHM methods in the fact that it uses spectral representation of the data. However, its implementation is different the FRF between two stmcturally mounted piezo wafers is determined directly through sweep-sine or broadband random excitation. The complex quantity measured in this way is also known as transmittance. FRF SHM methods can be either model based or model free. 

H(jf). which is the transfer function between sine wave variations of low amplitude of these quantities, has been calculated for the rotating disk system [10]. 

The application of a sine wave excitation to a system under test often is the easiest method of determining the system transfer function. Here we are concerned with measuring or inferring a transfer function for an electrochemical ceU as a first... 

ABSTRACT Voltammetric and thermoelectrochemical (TEC) transfer function measurements have been carried out to study the eleetrodeposition of silver from nitric and tartaric solutions. For an isothermal cell, the observed increase of the limiting current is due to the diffusion coefficient increase and to the mass transport boundary layer decrease when bath temperature increases. In a non-isothermal cell, through the use of sine wave temperature modulation, the TEC transfer function measurements show a typical mass transport responses and typical adsorption relaxation in middle frequency domain. The experimental data are in good accordance with previously developed model and permit to determine the diffusion activation energy and the densification coefficients of silver ions in this media. 

In previous works we have shown the great importance to control the working electrode temperature (non-isothermal process) and developed a new transfer function based on the sine wave modulation of the electrode temperature, (Olivier, Merienne, Chopart, and Aaboubi 1992). Thus the thermoelectrochemical (TEC) transfer function has been experimentally measured and compared with theoretical models for mass transport controlled systems or charge... 

The EIS technique is based on a transient response of an equivalent circuit for an electrode/solution interface. The response can be analyzed by transfer functions due to an applied smaU-amphtude potential excitation at varying signals or sweep rates, hi turn, the potential excitation yields current response and vice verse, hi impedance methods, a sine-wave perturbation of small amphtude is employed on a corroding system being modeled as an equivalent circuit (Figure 3.8) for determining the corrosion mechanism and the polarization resistance. Thus, a complex transfer function takes the form... 

Transfer functions can be used conveniently to obtain output responses to any type of input change. In this chapter we have focused on first- or second-order transfer functions and integrating processes. Because a relatively small number of input changes have industrial or analytical significance, we have considered in detail the responses of these basic process transfer functions to the important types of inputs, such as step, ramp, impulse, and sine inputs. 

We start with the response properties of a first-order process when forced by a sinusoidal input and show how the output response characteristics depend on the frequency of the input signal. This is the origin of the term frequency response. The responses for first- and second-order processes forced by a sinusoidal input were presented in Chapter 5. Recall that these responses consisted of sine, cosine, and exponential terms. Specifically, for a first-order transfer function with gain K and time constant t, the response to a general sinusoidal input, x t) = A sin (o, is... 

---