Complex transfer function

A complex transfer function may consist of processes in series, as shown in Fig. 2.25. This then gives the form... 

Figure 2.26. Time domain representation of complex transfer functions.

Simulation example TRANSIM is based on the solution of a complex transfer function. 

The Nyquist plot discussed in the previous section presents all the frequency information in a compact, one-curve form. Bode plots require that two curves be plotted instead of one. This increase is well worth the trouble because complex transfer functions can be handled much more easily using Bode plots. The two curves show how magnitude ratio and phase angle (argument) vary with frequency. 

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. 

Therefore the log modulus curves and phase-angle curves of the individual components arc simply added at each value of frequency to get the total L and d curves for the complex transfer function. 

Measurements of either r or at a particular frequency u can establish the value of T. With a more complex transfer function, two parameters can be established by such measurements. 

The spectroscopist should also pay attention to the question whether the sampling interval Js, the speed v of the movable mirror and the time constant T of the electronic data recording system are in a proper relation with each other. It is well known (see also Section 5.4) that a low pass filter with time constant t has the complex transfer function (amplitude and phase )... 

Although these relations will be written below for impedances they also hold for admittances and other complex transfer functions. Assuming that all the aforementioned conditions are met, the Kramers-KrcMiig relations are obtained allowing the calculation of the imaginary impedance from the real part... 

Several other forms of these relations can be found in the literature [3]. Although the preceding equations were written for impedances, they are valid for any complex transfer function. Kramers-Kronig relations are very restrictive, and in EIS some of them might be slightly relaxed, and instead of the impedances, the admittances can be used. This will be discussed in what follows. 

The cross-correlation of the input wave signal and the measured displacements of the paddle provides a complex transfer function that can be used to generate a wave board control signal that can ensure the desired waves. 

Fig. 40.5. Complex transfer function obtained by dynamic calibration.

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... 

In Chapter 5 we discussed the dynamics of relatively simple processes, those that can be modeled as either first- or second-order transfer functions or as an integrator. Now we consider more complex transfer function models that include additional time constants in the denominator and/or functions of s in the numerator. We show that the forms of the numerator and denominator of the transfer function model influence the dynamic behavior of the process. We also introduce a very important concept, the time delay, and consider the approximation of comphcated transfer function models by simpler, low-order models. Additional topics in this chapter include interacting processes, state-space models, and processes with multiple inputs and outputs. 

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... 

Here the output voltage and input displacement Xi are written in uppercase to emphasize that this transfer function operates in the frequency domain. It is a property of the Laplace transform that multiplication by the (complex) frequency s in the frequency domain is equivalent to differentiation in the time domain, so sX, is the input velocity. Another property of the Laplace transform is that to evaluate the (complex) transfer function at a given frequency f in Hz or angular frequency co = 2nf in rad/s, one makes the substitution s = jm. 

The modulus of the complex transfer function of an electrodynamic geophone with respect to grotmd velocity depends on damping as follows (Bormann 2012, Chapter 5.2) ... 

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