Bode plot

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. 

Then semilog graph paper can be used to plot both phase angle and log modulus versus the log of frequency, as shown in Fig. 12.12. There are very practical reasons for using these kinds of graphs, as we will find out shortly. 

Now let us look at the Bode plots of some common transfer functions. We have already calculated the magnitudes and phase angles for most of them in the previous section. The job now is to plot them in this new coordinate system. 

If G ) is just a constant K, G,j ) = K, and phase angle = arg G(Ib) = 0- Neither magnitude nor phase angle vary with frequency. The log modulus is 

Bode plots of phase angle and log modulus versus the logarithm of frequency. 

At low frequencies the capacitor is an open circuit and V0 should equal Vi. At high frequencies the capacitor becomes a short, and the gain goes toward zero. The 3 dB frequency of the circuit is to = 1/RC = 1,000 rad/s = 159 Hz. We will set up an AC Sweep to sweep the frequency from 1 Hz to 10 kHz. Select PSplce and then New Simulation Profile from the Capture menus and then enter a name for the profile and click the Create button. Select the AC Sweep/Nolse Analysis type and fill in the parameters as shown in the AC Sweep dialog box below  

Click the OK to accept the settings. Run the analysis by selecting PSpice and then Run from the Capture menus  

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))  

IK K Hvfiabtw Irtwu Fuflbtl AVOO AV0X(.) l 009(1 ENVM X(.) CNVMK) EXPt) 0 ) IM0() WOO LOOtO() MO l MAK[). j 

The DB command takes 201ogjo of the specified voltage. Thus, DB(V(VO)) is equivalent to 201og(V(Vo)). Since our source voltage (Vi) had a magnitude of 1 V, DB(V(VO is equivalent to DB(V(Vo)/Vl), which is the Bode plot of the gain of this circuit. Click the OK button to plot the trace  

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First one assumes that the final closed loop compensation network will have a continuous -20dB/decade slope. To achieve a 15 kHz cross-over frequency, the amplifier must add gain to the input signal and push-up the gain curve of the Bode plot. 

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).

The resulting gain and phase Bode plots ean be seen in Figure 4-24. 

Some implementations of these error amp eireuits are shown in Figures B-5 through B-7. Some useful mathematieal tools when working with Bode plots are given below. 

Gdc is then the starting point of the gain Bode plot at dc. 

The resulting control-to-output Bode plots for the voltage-mode controlled forward converter are given in Figure B-11. 

The schematic and Bode plot for the single-pole method of compensation are given in Figure B-16. At dc it exhibits the full open-loop gain of the op amp, and its gain drops at -20dB/decade from dc. It also has a constant -270 degree phase shift. Any phase shift contributed by the control-to-output characteristic... 

Sinee 1 / 2 is —3 dB, the exaet modulus passes 3 dB below the asymptote interseetion at /T rad/s. The asymptotie eonstruetion of the log modulus Bode plot for a first-order system is shown in Figure 6.8. 

The singular value of the sensitivity funetion irfSfja )) and of the eomplementary sensitivity funetion (r(T(ja )) ean be displayed as Bode plots and play an important role in robust multivariable eontrol system design. 

Bode Plot a graph of the frequency response see Frequency Response Analysis) of an electrode whereby the magnitude and the phase angle are separately plotted as a function of the frequency. 

The technique of AC Impedance Spectroscopy is one of the most commonly used techniques in electrochemistry, both aqueous and solid.49 A small amplitude AC voltage of frequency f is applied between the working and reference electrode, superimposed to the catalyst potential Uwr, and both the real (ZRe) and imaginary (Zim) part of the impedance Z (=dUwR/dI=ZRc+iZim)9 10 are obtained as a function of f (Bode plot, Fig. 5.29a). Upon crossplotting Z m vs ZRe, a Nyquist plot is obtained (Fig. 5.29b). One can also obtain Nyquist plots for various imposed Uwr values as shown in subsequent figures. 

Figure 17. EIS Bode plot for epoxy coating over zinc-primed steel exposed for 2 months to uninoculated medium (curve 1) and six mixed cultures of facultative anaerobes (curves 2-7). (Reprinted from Ref 35 with permission from NACE International.)...

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. 

A preview We can derive the ultimate gain and ultimate period (or frequency) with stability analyses. In Chapter 7, we use the substitution s = jco in the closed-loop characteristic equation. In Chapter 8, we make use of what is called the Nyquist stability criterion and Bode plots. 

Nyquist plot Bode plot Nyquist plot is a frequency parametric plot of the magnitude and the argument of the open-loop transfer function in polar coordinates. Bode plot is magnitude vs. frequency and phase angle vs. frequency plotted individually. 

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. 

Another advantage of frequency response analysis is that one can identify the process transfer function with experimental data. With either a frequency response experiment or a pulse experiment with proper Fourier transform, one can construct the Bode plot using the open-loop transfer functions and use the plot as the basis for controller design.1... 

This plotting format contains the same information as the Bode plot. The polar plot is more compact, but the information on the frequency is not shown explicitly. If we do not have a computer, we theoretically could read numbers off a Bode plot to construct the Nyquist plot. The use of Nyquist plots is more common in multiloop or multivariable analyses. A Bode plot, on the... 

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

There are two possibilities. They are shown in Fig. 8.4, together with their interpretations on a Bode plot. 

The Nyquist stability criterion can be applied to Bode plots. In fact, the calculation using the Bode plot is much easier. To obtain the gain margin, we find the value of GCGP which corresponds to a phase lag of-180°. To find the phase margin, we look up the phase lag corresponding to when GCGP is 1. 

First, we need to generate the plots. Use Fig. E8.12 to help interpret the MATLAB generated Bode plot.1... 

On the Bode plot, the comer frequencies are, in increasing order, l/xp, Zq, and p0. The frequency asymptotes meeting at co = l/xp and p0 are those of a first-order lag. The frequency asymptotes meeting at co = z0 are those of a first-order lead. The largest phase lag of the system is -90° at very high frequencies. The system is always stable as displayed by the root locus plot. 

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... 

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