Generalized Transfer Functions

We now generalize our simple illustration. Consider a general transfer function of a stable model G(s), which we also denote as the ratio of two polynomials, G(s) = Q(s)/P(s). We impose a sinusoidal input f(t) = A sin cot such that the output is... 

G. GENERAL TRANSFER FUNCTIONS IN SERIES. The historical reason for the widespread use of Bode plots is that, before the use of computers, they made it possible to handle complex processes fairly easily. A complex transfer function can be broken down into its simple elements leads, lags, gains, deadtimes, etc. Then each of these is plotted on the same Bode plots. Finally the total complex transfer function is obtained by adding the individual log modulus curves and the individual phase curves at each value of frequency. 

Consider a general transfer function G,j) that can be broken up into two simple transfer functions and G u, ... 

The tanks-in-series model is a flexible one-parameter model which amounts to characterising a system in terms of the general transfer function of the equation... 

For symmetric effective coupling topogolies with = J23, the general transfer function T 2 of Eq. (193) can be reduced to... 

The methods described in this chapter and this book apply to electrochemical impedance spectroscopy. Impedance spectroscopy should be viewed as being a specialized case of a transfer-function analysis. The principles apply to a wide variety of frequency-domain measurements, including non-electrochemical measurements. The application to generalized transfer-function methods is described briefly with an introduction to other sections of the text where these methods are described in greater detail. Local impedance spectroscopy, a relatively new and powerful electrochemical approach, is described in detail. 

While the emphasis of this book is on electrochemical impedance spectroscopy, the methods described in Section 7.3 for converting time-domain signals to frequency-domain transfer functions clearly are general and can be applied to any t3q>e of input and output. Some generalized transfer-function approaches are described in Chapters 14 and 15. 

Table 7.1 Generalized transfer functions for a rotating disk electrode at fixed temperature.

Four state variables may be defined, for example, for the rotating disk described in Qiapter 11. These may include the rotation speed, the temperature, the current, and the potential. At a fixed temperature, three variables remain from which a transfer function may be calculated. As shown in Table 7.1, the generalized transfer functions include impedance, admittance (see Chapter 16), and two types of electrohydrodynamic impedance (see Chapter 15). 

Local impedance measurements represent another form of generalized transfer-function analysis. In these experiments, a small probe is placed near tiie electrode surface. The probe uses either two small electrodes or a vibrating wire to allow measurement of potential at two positions. Under the assumption that the electrolyte conductivity between the two points of potential measurement is uniform, the current density at the probe can be estimated from the measured potential difference AVprobe by... 

The development presented here for the complex impedance, Z = Zr+ jZj, is general and can be applied, for example, to the complex refractive index, the complex viscosity, and the complex permittivity. The derivation for a general transfer function G follows that presented by Nussenzveig. The development for the subsequent analysis in terms of impedance follows the approach presented by Bode. ... 

This part demonstrates how deterministic models of impedance response can be developed from physical and kinetic descriptions. When possible, correspondence is drawn between hypothesized models and electrical circuit analogues. The treatment includes electrode kinetics, mass transfer, solid-state systems, time-constant dispersion, models accounting for two- and three-dimensional interfaces, generalized transfer functions, and a more specific example of a transfer-function tech-nique.in which the rotation speed of a disk electrode is modulated. 

Note that this is analogous to the time-domain version of a general transfer function f(t)... 

On the other hand, by applying multidimensional Fourier transform on the function gnin,..., Tn), the nth-order FRF or the nth-order generalized transfer function is obtained ... 

Finally, the (Tn, T12, T21 and T22) parameters allow us to calculate two important quantities the generalized transfer function can be represented as ratio of Vi to Vs, and the input impedance of the network with distributed branches. 

By using the multiphcation and division properties, the amplitude ratio and phase angle of the above mentioned general transfer function, consisting of multiple terms, becomes ... 

From this example, we conclude that direct analysis of the complex transfer function G( (o) is computationally easier than solving for the actual long-time output response. The computational advantages are even greater when dealing with more complicated processes, as shown in the following. Start with a general transfer function in factored form... 

A general transfer function for a second-order system without numerator dynamics is... 

The Modulation Transfer Function is one part of the more general Transfer Function - it is the part that deals with the magnitude or contrast of the output as a fraction of the magnitude of the input. It is generally the real component of the TF. Generally we normalize the MTF so that it is unity at zero frequency. 

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