First-order systems Nyquist plot

Construct the Bode diagram and Nyquist plot of a first-order system with dead time, having a transfer function... 

For a first order function with deadtime, the proportional gain, integral and derivative time constants of an ideal PID controller. Can handle dead-time easily and rigorously. The Nyquist criterion allows the use of open-loop functions in Nyquist or Bode plots to analyze the closed-loop problem. The stability criteria have no use for simple first and second order systems with no positive open-loop zeros. 

Figure 17.14 Nyquist plots for various systems (a) first-order (b) second-order (c) third-order (d) pure dead-time (e) pure capacitive.

To generate the HGAf(jw) Nyquist plots discussed above, the Z transform of the appropriate transfer functions must first be obtained. Then is substituted for Z, and (i) is varied from 0 to o)j/2. There is an alternative method that is often more convenient to use, particularly in high-order systems. Equation (15.40) gives a doubly irifinite series representation of HGM m)-... 

The frequency response of sampled-data systems can be easily calculated using MATLAB software. Table 15.2 gives a program that generates a Nyquist plot for the first-order process with a deadtfme of one sampling period. 

In order to correctly interpret the experimental information provided by EIS, conveyed in either Nyquist or Bode plots, the use of a sound physical model describing the relevant biophysicochemical processes taking place in the system is essential. A simple strategy to deal with the experimental information involves the implementation of the model into an equivalent circuit, which contains all the information of charge transport. In the equivalent circuit, the resistances and capacitances describe the charge loss and accumulation mechanisms that can take place in the system. In the following section, we first describe the most common circuit elements used in EIS data analysis, followed by the most common equivalent circuits used to describe typical electrochemical cells. 

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