Frequency response analysis

Consider a first-order process with transfer function  

Let us assume that this process is subjected to a sinusoidal input change  

Using Laplace transform tables, Eqn. (9.2) can be transformed to the Laplace domain  

Note thatX S ) is defined as L[Sy(t)], the Laplace transform of (f) Eqnation (9.4) ean be expanded into fractions and therefore be written as  

The constants B can now be calculated and using tables of inverse Laplace transform, the solntion for t — o can be written as  

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Cyclic voltammetry (adsorption, monolayers) Potentiodynamic polarisation (passivation, activation) Cathodic reduction (thickness) Frequency response analysis (electrical properties, heterogeneity) Chronopotentiometry (kinetics)... 

Electrical characteristics of surface films formed electrochemically can be analysed using frequency response analysis (FRA) (sometimes called electrochemical impedance spectroscopy, or This technique is... 

Bode Plot a graph of the frequency response see Frequency Response Analysis) of an electrode whereby the magnitude and the phase angle are separately plotted as a function of the frequency. 

Electrochemical Impedance Spectroscopy see Frequency Response Analysis. 

Frequency Response Analysis the response of an electrode to an imposed alternating voltage or current sign of small amplitude, measured as a function of the frequency of the perturbation. Also called Electrochemical Impedance Spectroscopy. 

Sodium dodecyl sulfate has been used to modify polypyrrole film electrodes. Electrodes synthesized in the presence of sodium dodecyl sulfate have improved redox processes which are faster and more reversible than those prepared without this surfactant. The electrochemical behavior of these electrodes was investigated by cyclic voltametry and frequence response analysis. The electrodes used in lithium/organic electrolyte batteries show improved performance [195]. 

We have given up the pretense that we can cover controller design and still have time to do all the plots manually. We rely on MATLAB to construct the plots. For example, we take a unique approach to root locus plots. We do not ignore it like some texts do, but we also do not go into the hand sketching details. The same can be said with frequency response analysis. On the whole, we use root locus and Bode plots as computational and pedagogical tools in ways that can help to understand the choice of different controller designs. Exercises that may help such thinking are in the MATLAB tutorials and homework problems. 

In effect, we are adding a very large real pole to the derivative transfer function. Later, after learning root locus and frequency response analysis, we can make more rational explanations, including why the function is called a lead-lag element. We ll see that this is a nice strategy which is preferable to using the ideal PD controller. 

The literature refers the term as a first order filter. It only makes sense if you recall your linear circuit analysis or if you wait until the chapter on frequency response analysis. 

One may question whether direct substitution is a better method. There is no clear-cut winner here. By and large, we are less prone to making algebraic errors when we apply the Routh-Hurwitz recipe, and the interpretation of the results is more straightforward. With direct substitution, we do not have to remember ary formulas, and we can find the ultimate frequency, which however, can be obtained with a root locus plot or frequency response analysis—techniques that we will cover later. 

Note 1 This result is consistent with the use of frequency response analysis later in Chapter 8. 

In this computer age, one may question why nobody would write a program that can solve for the roots with dead time accurately Someone did. There are even refined hand sketching techniques to account for the lag due to dead time. However, these tools are not as easy to apply and are rarely used. Few people use them because frequency response analysis in Chapter 8 can handle dead time accurately and extremely easily. 

On the other hand, frequency response analysis cannot reveal information on dynamic response easily—something root locus does very well. Hence controller design is always an iterative procedure. There is no one-stop-shopping. There is never a unique answer. 

Frequency response analysis allows us to derive a general relative stability criterion that can easily handle systems with time delay. This property is used in controller design. 

This is a crucial result. It constitutes the basis of frequency response analysis, where in general, all we need are the magnitude and the argument of the transfer function G(s) after the substitution s = jco. 

We need to appreciate some basic properties of transfer functions when viewed as complex variables. They are important in performing frequency response analysis. Consider that any given... 

We do not need to expand the entire function into partial fractions. The functions Gb G2, etc., are better viewed as simply first and at the most second order functions. In frequency response analysis, we make the s = jto substitution and further write the function in terms of magnitude and phase angle as ... 

With these results, we are ready to construct plots used in frequency response analysis. The important message is that we can add up the contributions of individual terms to construct the final curve. The magnitude, of course, would be on the logarithmic scale. 

Another advantage of frequency response analysis is that one can identify the process transfer function with experimental data. With either a frequency response experiment or a pulse experiment with proper Fourier transform, one can construct the Bode plot using the open-loop transfer functions and use the plot as the basis for controller design.1... 

The pulse experiment is not crucial for our understanding of frequency response analysis and is provided on our Web Support, but we will do the design calculations in Section 8.3. 

This example shows us the very important reason why and how frequency response analysis can... 

With frequency response analysis, we can derive a general relative stability criterion. The result is apphcable to systems with dead time. The analysis of the closed-loop system can be reduced to using only the open-loop transfer functions in the computation. 

The gain and phase margins are used in the next section for controller design. Before that, let s plot different controller transfer functions and infer their properties in frequency response analysis. Generally speaking, any function that introduces additional phase lag or magnitude tends to be destabilizing, and the effect is frequency dependent. 

We know from frequency response analysis that time lag introduces extra phase lag, reduces the gain margin and is a significant source of instability. This is mainly because the feedback information is outdated. 

When we use pade () without the left-hand argument [q, p], the function automatically plots the step and phase responses and compares them with the exact responses of the time delay. Pade approximation has unit gain at all frequencies. These points will not make sense until we get to frequency response analysis in Chapter 8. So for now, keep the [q, p] on the left hand side of the command. 

After that, we can use the result to do frequency response analysis. If you are interested in the details, they are provided in the Session 7 Supplement on our Web Support. 

As is well known in the field of electrochemistry in general, electrode kinetics may be conveniently examined by cyclic voltammetry (CV) and by frequency response analysis (ac impedance). The kinetics of the various polymer electrodes considered so far in this chapter will be discussed in terms of results obtained by these two experimental techniques. 

These studies have been mainly carried out using cyclic voltammetry and frequency response analysis as experimental tools. As a typical example. Fig. 9.12 illustrates the voltammogram related to the p-doping process of a polypyrrole film electrode in the LiClQ -propylene carbonate electrolyte, i.e. the reaction already indicated by (9.16). 

Further information on this subject can be obtained by frequency response analysis and this technique has proved to be very valuable for studying the kinetics of polymer electrodes. Initially, it has been shown that the overall impedance response of polymer electrodes generally resembles that of intercalation electrodes, such as TiS2 and WO3 (Ho, Raistrick and Huggins, 1980 Naoi, Ueyama, Osaka and Smyrl, 1990). On the other hand this was to be expected since polymer and intercalation electrodes both undergo somewhat similar electrochemical redox reactions, which include the diffusion of ions in the bulk of the host structures. One aspect of this conclusion is that the impedance response of polymer electrodes may be interpreted on the basis of electrical circuits which are representative of the intercalation electrodes, such as the Randles circuit illustrated in Fig. 9.13. The figure also illustrates the idealised response of this circuit in the complex impedance jZ"-Z ) plane. 

J. W. Blackburn. 1989. Frequency response analysis of naphthalene biotransformation activity. Biochemical Engineering VI, Aimals N. Y. Acad. Scien. In Press. 

Frequency response analysis of transducer used to measure pressure in the closed bomb system) 68)K.K.Andreev A.F. 

A cyclic input of tracer will give rise to a cyclic output. This type of input is more troublesome to apply than a pulse or step input but has some advantages for frequency response analysis. A truly instantaneous pulse input is also difficult to apply in practice, but an idealised input pulse is not essential 2,3 for obtaining the desired information (See Section 2.3.5 (b)). 

Kramers, H. and Alberda, G. Chem. Eng. Sci. 2 (1953) 173. Frequency-response analysis of continuous-flow systems. 

If xj/ < 0 the output is said to lag the input, and if xj/ > 0 it is said to lead the input. The relationship between input and output for a sinusoidal forcing function as t -> oo constitutes an important tool in the analysis and design of control systems termed frequency response analysis. Of particular importance are the amplitude ratio (AR) and the phase shift xfr. The AR represents the relationship between the output and the input amplitudes... 

C. Gabrielli, Identification of Electrochemical Processes by Frequency Response Analysis, Techn. report n° 004/83. Schlumberger. Farnborough (1980). 

C. Gabrielli, The identification of electrochemical processes by frequency response analysis, Solartron Instruments, UK, 1980. 

The maximum of the semicircle appears at wmax = kiec + k . It follows that frequency response analysis of the small signal photopotential response should give access not only to klr and krec but also to Csc and CH. The possibilities offered by intensity modulated photopotential measurements remain largely unexplored, although Kamieniecki used the method... 

Light and potential modulated microwave reflectivity measurements offer a novel approach to the study of the semiconductor electrolyte interface. Perturbation of the density of electrons and holes in a semiconductor influences the conductivity and hence the imaginary component of the dielectric constant at microwave frequencies. For small perturbations, the change ARm in microwave reflectivity depends linearly on the change in conductivity [27, 28, 75). The application of frequency response analysis to light modulated microwave reflectance is relatively new [30]. Although the technique is analogous to IMPS, it provides additional information. 

In order to actually cover 19 decades in frequency, dielectric spectroscopy makes use of different measurement techniques each working at its optimum in a particular frequency range. The techniques most commonly applied include time-domain spectroscopy, frequency response analysis, coaxial reflection and transmission methods, and at the highest frequencies quasi-optical and Fourier transform infrared spectroscopy (cf. Fig. 2). A detailed review of these techniques can be found in Kremer and Schonhals [37] and in Lunkenheimer [45], so that in the present context only a few aspects will be summarized. 

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