Nyquist

Helpful documentation for both Nyquist and XLISP are given in the Nyquist Reference Manual in folder nyquist. Also, good introductory articles and a short tutorial can be found in Computer Music Journal, 21 3, published in 1997. 

Lisp presents itself to the user as an interpreter it works both as a programming language and as an interactive system it waits for some input or command from the user, executes it, and waits again for further input. Lisp is an acronym for List Processor almost everything in the Lisp world is a list. In programming terms, a list is a set of elements enclosed between parentheses, separated by spaces. The elements of a list can be numbers, symbols, other lists, and indeed, programs. Examples of lists are  

When the interpreter computes (or evaluates, in Lisp jargon) a list, it always assumes that the first element of the list is the name of a function and the rest of the elements are the arguments that the function needs for processing. In the above example, the interpreter performs the addition of two numbers the name of the function is the symbol + and the two arguments are the numbers 330 and 336. 

If an argument for a function is also a function itself, then the argument is evaluated first, unless specified otherwise. The general rule is that the innermost function is evaluated first. In the case of various nested functions, the outermost one is normally the last to get evaluated. The Lisp interpreter treats each nested list or function independently. For example  

In this example, the function + is evoked with number 330 as the first argument and with the result of the evaluation of ( 56 (+ 5 1)) as the second ailment, which in turn evokes the function with number 56 as the first argument and with the result of the evaluation of (+ 5 1) as the second arguments. The result of the innermost step is 6, which is then multiplied by 56. The result of the multiplication is 336 which is finally added to 330. The result of the whole function therefore is 6.  

After letting the experiment last for a reasonable duration, it is sometimes possible to reach a current/voltage point that does not change with time. In this case one refers to a steady or stationary state. The steady (U,I) or ( )/) curves that are plotted in these conditions are useful tools for analysing the behaviour of electrochemical systems .  

This notion must not be confused with the notion of equilibrium in other words, the terms steady and constant are not interchangeable. For a steady state to be established, it is merely necessary that the relevant parameters, at any point in space, do not vary with time. Strictly speaking, an electrochemical system in a steady state can therefore present spatial gradients, but the latter must not vary with time. For instance, as far as the concentrations are concerned, the steady state is expressed by the following  

Equilibrium states are particular cases of steady states in which the overall current is 

From an experimental point of view, what is frequently observed are (U,I) or (E,I) curves which are steady throughout the duration of the experiment. But the steady character of the various parameters, such as the concentrations in the electrolyte, is not necessarily strictly verified. Therefore one should use the more accurate term of quasisteady state. It is very exceptional in electrochemistry to find strict steady states differing from equilibrium states. This is because no time evolution of the mean composition of each phase must then occur. The systems showing steady (U,I) or E,I) characteristics throughout the duration of the experiment are thus therefore most of the time in a quasi-steady state. 

The first precondition for securing a quasi-steady state is to ensure that the system s overall chemical composition be kept practically unchanged from start to end of the experiment. 

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Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

Nyquist theorem statement that a periodic signal must be... 

Nyquist criterion Nyquist theorem Nystan Nystarescent Nystatin... 

The Nyquist critical frequency or critical angular frequency is... 

Vibration analysis This ineludes an on-line analysis of the vibration signals, FFT speetral analysis, transient analysis, and diagnosties. A wide variety of displays are available ineluding orbits, easeades, bode and nyquist plots, and transient plots. 

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then... 

In order to eneirele any poles or zeros of F s) that lie in the right-hand side of the. v-plane, a Nyquist eontour is eonstrueted as shown in Figure 6.16. To avoid poles at the origin, a small semieirele of radius e, where e 0, is ineluded. 

The Nyquist stability criterion can be stated as A closed-loop control system is stable if, and only if, a contour in the G s)H s) plane describes a number of counterclockwise encirclements of the (—l,jO) point, the number of encirclements being equal to the number of poles of G s)H s) with positive real parts . 

In practice, only the frequencies lu = 0 to+oo are of interest and since in the frequency domain. v = jtu, a simplified Nyquist stability criterion, as shown in Figure 6.18 is A closed-loop system is stable if, and only if, the locus of the G(iLu)H(iuj) function does not enclose the (—l,j0) point as lu is varied from zero to infinity. Enclosing the (—1, jO) point may be interpreted as passing to the left of the point . The G(iLu)H(iLu) locus is referred to as the Nyquist Diagram. 

Fig. 6.18 Nyquist diagram showing stable and unstable contours.

Construet the Nyquist diagram for the eontrol system shown in Figure 6.20 and find the eontroller gain K that... 

Table 6.3 Data for Nyquist diagram for system in Figure 6.20...

Then n in equation (6.62) is the type number of the system and J([ denotes the product of the factors. The system type can be observed from the starting point uj 0) of the Nyquist diagram, and the system order from the finishing point bj oo), see Figure 6.22. 

Fig. 6.22 Relationship between system type classification and the Nyquist diagram. For a step input,...

The M and N circles can be superimposed on a Nyquist diagram (called a Hall chart) to directly obtain closed-loop frequency response information. 

Alternatively, the closed-loop frequency response can be obtained from a Nyquist diagram using the direct construction method shown in Figure 6.25. From equation (6.73)... 

Fig. 6.25 Closed-loop frequency response from Nyquist diagram using the direct construction method.

Shannon s sampling theorem states that A funetion f t) that has a bandwidth is uniquely determined by a diserete set of sample values provided that the sampling frequeney is greater than 2uj, . The sampling frequeney 2tJb is ealled the Nyquist frequeney. 

Robust stability can be investigated in the frequency domain, using the Nyquist stability criterion, defined in section 6.4.2. 

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