Bode plots examples

Fig. 4.12 Absolute value of impedance and phase angle vs. frequency (Bode plot). Example p-GaAs in 7 mM Cu aqueous solution at 0.4 V vs. saturated calomel electrode (SCE) the solid curve is calculated from a equivalent circuit, the squares and circles are experimental data. (After ref. (35]).

There are other kinds of graphing AC impedance data, useful when experimental data do not form exactly a semi-circle. It is possible to plot Z versus co, and Z" versus o) (Bode plots). Examples of these curves are visible in Figures 10.9 and 10.10. 

Figure 3-64 The gain and phase Bode plots for the design example 3.15.1 (a) the phase plot for the buck converter (b) the phase plot for the buck converter.

Figure 3-67 The gain and phase Bode plots for design example 3.15.2 (compensation design).

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. 

We know that both G(jco) and Z(G(jco)) are functions of frequency, co. We certainly would like to see the relationships graphically. There are three common graphical representations of the frequency dependence. We first describe all three methods briefly. Our introduction relies on the use the so-called Bode plots and more details will follow with respective examples. 

Even with MATLAB, we should still know the expected shape of the curves and its telltale features. This understanding is crucial in developing our problem solving skills. Thus doing a few simple hand constructions is very instructive. When we sketch the Bode plot, we must identify the comer (break) frequencies, slopes of the magnitude asymptotes and the contributions of phase lags at small and large frequencies. We ll pick up the details in the examples. 

Example 8.9. Sketch the Bode plot of the following transfer function ... 

This is a big question when we use, for example, a Bode plot. Let s presume that we have a closed-loop system in which we know "everything" but the proportional gain (Fig. 8.5), and we write the closed-loop characteristic equation as... 

How do I know the answer is correct Just "plug" Kc back into G0l and repeat the Bode plot using G0l It does not take that much time to check with MATLAB. Now, we are finally ready for some examples. Again, run MATLAB to confirm the results while you read them. 

Before we give some examples of the design of feedback controllers in the frequency domain, it would 1m wise to show what the common P, PI, and PID controllers look like in the frequency domain. These will be the that we will add to the process to get the total openloop Bode plots of. ... 

At this point it might be useful to pull together some of the concepts that you have waded through in the last several chapters. We now know how to look at and think about dynamics in three languages time (English), Laplace (Russian) and frequency (Chinese). For example, a third-order, underdamped system would have the time-domain step responses sketched in Fig. 14.10 for two different values of the real TOOt. In the Laplace domain, the system is represented by a transfer function or by plotting the poles of the transfer function (the roots of the system s characteristic equation) in the s plane, as shown in Fig. 14.10. In the frequency domain, the system could be represented by a Bode plot of... 

The phase and magnitude can be displayed as shown in previous sections. In this example we would like to show a Bode plot that displays the magnitude in decibels. To display Vo in decibels, we need to display the trace dB(V(VO)). Select Trace and then Add Trace from the Probe menu bar. In the text field next to Trace Expression enter the text DB (V(VO)) ... 

The plot shows four Bode plots, one for each value of RF V3l. We see that for larger gains the bandwidth is reduced. This example will be continued in the next section (Section 5.G), which starts with the screen capture above. You may wish to keep the Probe window above open so that you can easily continue with the next section. 

Different kinds of plots based on impedance Z, admittance Z 1, modulus icoZ, or complex capacitance (z coZ) 1 can be used to display impedance data. In solid state ionics, particularly plots in the complex impedance plane (real versus imaginary part of Z) and impedance Bode-plots (log(Z) log(co)) are common. A RC element (resistor in parallel with a capacitor) has, for example, an impedance according to... 

IMPS data are usually fitted in the complex plane, but Bode plots of the magnitude and phase of the IMPS response offer a more sensitive way of deconvoluting RC attenuation effects from the kinetics of carrier transport and recombination. An example of the comparison between theory and experiment in Bode plot format is shown in Fig. 8.32. 

EIS can also detect defects arising from lack of adhesion at adhesively bonded surfaces (111). The presence of such defects produces pronounced changes in the character of the data presented either in complex plane plots or in Bode plots. Figure 38 illustrates the measurement configuration and provides examples of EIS data for defective and defect-free samples. Studies have shown that the presence of defects is readily revealed and that the geometry of the defects and their spatial extent can be inferred from a detailed analysis of EIS spectra. 

Figure 2.37 shows an example impedance spectrum of an electrochemical system with two time constants. Figure 2.37a, b, and c are the equivalent circuit, simulated Nyquist diagram, and Bode plot, respectively. 

Experimental arcs in the spectrum are not always ideal semicircles, and this complicates parameter estimation. Nevertheless, there are still basic rules for estimating the initial values [8, 9], The key is to identify the region of the spectrum in which one element dominates and then estimate the value of the element in this region. For example, the resistor s impedance dominates the spectrum at a low frequency, while the impedance of a capacitor approaches zero at a high frequency and infinity at a low frequency also, individual resistors can be recognized based on the horizontal regions in a Bode plot. 

Another method often used for plotting and evaluating EIS data involves plots of log Z and 8 versus log GO. These data presentations are known as Bode plots and are illustrated by the example in Fig. 6.20, again for the simplest equivalent circuit of Fig. 6.18. Bode plots have advantages in that the impedance and impedance phase angle are shown as explicit functions of the frequency, which is the independent experimental variable. Reference to Eq 6.68 shows that at very high go values, Z approaches Rs, and at very low frequencies, Z approaches (Rs + Rp). These limits are indicated in Fig. 6.20. In analyzing intermediate frequencies, that is, when (Rs + Rp) > Z ... 

The connection between the impedance and dielectric measurements can be seen easily for only relatively simple examples. In Figure 7-8c the complex capacitance plot for the same equivalent circuit is similar to typical dielectric response dielectric only in the low-frequency region. The plot of the magnitude of the complex impedance, also shown in this panel, is known as the Bode plot. 

It has become common practice to fit IMPS data in the complex plane, but this approach is rather unsatisfactory when the effects of RC attenuation are convoluted with those due to carrier transport and recombination. A more sensitive method is to use Bode plots. The magnitude and phase angle plots provide an excellent diagnostic analysis. An example from the work of Dloczik et al. [90] is shown in Fig. 50. 

The Bode plots are easily constructed and shown in Figure 17.4. Second-order system In Example 17.3 we found that... 

Figure 17.7 Bode plots for two capacities in series (Example 17.6).

Example 17.7 Bode Plots for an Open-Loop System... 

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