Бөкей сызбасы: Жүйе жиілік жауабының графигі

Graph of the frequency response of a control system

In electrical engineering and control theory, a Bode plot '/b//oʊ//d//i/// is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude (usually in decibels) of the frequency response, and a Bode phase plot, expressing the phase shift. As originally conceived by Hendrik Wade Bode in the 1930s, the plot is an asymptotic approximation of the frequency response, using straight line segments.

Among his several important contributions to circuit theory and control theory, engineer Hendrik Wade Bode, while working at Bell Labs in the 1930s, devised a simple but accurate method for graphing gain and phase shift plots. These bear his name, Bode gain plot and Bode phase plot. "Bode" is often pronounced '/b//oʊ//d//i/ BOHdee, although the Dutch pronunciation is Bo duh. Bode was faced with the problem of designing stable amplifiers with feedback for use in telephone networks. He developed the graphical design technique of the Bode plots to show the gain margin and phase margin required to maintain stability under variations in circuit characteristics caused during manufacture or during operation. The principles developed were applied to design problems of servomechanisms and other feedback control systems. The Bode plot is an example of analysis in the frequency domain.

The Bode plot for a linear, time invariant system with transfer function ( being the complex frequency in the Laplace domain) consists of a magnitude plot and a phase plot. The Bode magnitude plot is the graph of the function of frequency (with being the imaginary unit). The axis of the magnitude plot is logarithmic and the magnitude is given in decibels, i. e., a value for the magnitude is plotted on the axis at
The Bode phase plot is the graph of the phase, commonly expressed in degrees, of the transfer function as a function of The phase is plotted on the same logarithmic axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.

The horizontal frequency axis, in both the magnitude and phase plots, can be replaced by the normalized (nondimensional) frequency ratio In such a case the plot is said to be normalized, and units of the frequencies are no longer used, since all input frequencies are now expressed as multiples of the cutoff frequency .

Figures 2 5 further illustrate construction of Bode plots. This example with both a pole and a zero shows how to use superposition. To begin, the components are presented separately. Figure 2 shows the Bode magnitude plot for a zero and a low pass pole, and compares the two with the Bode straight line plots. The straight line plots are horizontal up to the pole (zero) location and then drop (rise) at 20 dB/decade. The second Figure 3 does the same for the phase. The phase plots are horizontal up to a frequency factor of ten below the pole (zero) location and then drop (rise) at 45°/decade until the frequency is ten times higher than the pole (zero) location. The plots then are again horizontal at higher frequencies at a final, total phase change of 90°. Figure 4 and Figure 5 show how superposition (simple addition) of a pole and zero plot is done. The Bode straight line plots again are compared with the exact plots. The zero has been moved to higher frequency than the pole to make a more interesting example. Notice in Figure 4 that the 20 dB/decade drop of the pole is arrested by the 20 dB/decade rise of the zero resulting in a horizontal magnitude plot for frequencies above the zero location. Notice in Figure 5 in the phase plot that the straight line approximation is pretty approximate in the region where both pole and zero affect the phase. Notice also in Figure 5 that the range of frequencies where the phase changes in the straight line plot is limited to frequencies a factor of ten above and below the pole (zero) location. Where the phase of the pole and the zero both are present, the straight line phase plot is horizontal because the 45°/decade drop of the pole is arrested by the overlapping 45°/decade rise of the zero in the limited range of frequencies where both are active contributors to the phase.

Figures 6 and 7 illustrate the gain behavior and terminology. For a three pole amplifier, Figure 6 compares the Bode plot for the gain without feedback (the open loop gain) AOL with the gain with feedback AFB (the closed loop gain). See negative feedback amplifier for more detail. In this example, AOL = 100 dB at low frequencies, and 1 / β = 58 dB. At low frequencies, AFB ≈ 58 dB as well. Because the open loop gain AOL is plotted and not the product β AOL, the condition AOL = 1 / β decides f0 dB. The feedback gain at low frequencies and for large AOL is AFB ≈ 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain AOL), so an equivalent way to find f0 dB is to look where the feedback gain intersects the open loop gain. (Frequency f0 dB is needed later to find the phase margin.) Near this crossover of the two gains at f0 dB, the Barkhausen criteria are almost satisfied in this example, and the feedback amplifier exhibits a massive peak in gain (it would be infinity if β AOL = −1). Beyond the unity gain frequency f0 dB, the open loop gain is sufficiently small that AFB ≈ AOL (examine the formula at the beginning of this section for the case of small AOL). Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency f180 where the open loop gain has a phase of −180°. In this vicinity, the phase of the feedback amplifier plunges abruptly downward to become almost the same as the phase of the open loop amplifier. (Recall, AFB ≈ AOL for small AOL.) Comparing the labeled points in Figure 6 and Figure 7, it is seen that the unity gain frequency f0 dB and the phase flip frequency f180 are very nearly equal in this amplifier, f180 ≈ f0 dB ≈ 3.332 kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable. Figures 8 and 9 illustrate the gain margin and phase margin for a different amount of feedback β. The feedback factor is chosen smaller than in Figure 6 or 7, moving the condition | β AOL | = 1 to lower frequency. In this example, 1 / β = 77 dB, and at low frequencies AFB ≈ 77 dB as well. Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and AOL occurs at f0 dB = 1 kHz. Notice that the peak in the gain AFB near f0 dB is almost gone. Figure 9 is the phase plot. Using the value of f0 dB = 1 kHz found above from the magnitude plot of Figure 8, the open loop phase at f0 dB is −135°, which is a phase margin of 45° above −180°. Using Figure 9, for a phase of −180° the value of f180 = 3.332 kHz (the same result as found earlier, of course). The open loop gain from Figure 8 at f180 is 58 dB, and 1 / β = 77 dB, so the gain margin is 19 dB. Stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good step response. As a rule of thumb, good step response requires a phase margin of at least 45°, and often a margin of over 70° is advocated, particularly where component variation due to manufacturing tolerances is an issue. See also the discussion of phase margin in the step response article.

The Bode plotter is an electronic instrument resembling an oscilloscope, which produces a Bode diagram, or a graph, of a circuit's voltage gain or phase shift plotted against frequency in a feedback control system or a filter. An example of this is shown in Figure 10. It is extremely useful for analyzing and testing filters and the stability of feedback control systems, through the measurement of corner (cutoff) frequencies and gain and phase margins. This is identical to the function performed by a vector network analyzer, but the network analyzer is typically used at much higher frequencies. For education and research purposes, plotting Bode diagrams for given transfer functions facilitates better understanding and getting faster results (see external links).

Two related plots that display the same data in different coordinate systems are the Nyquist plot and the Nichols plot. These are parametric plots, with frequency as the input and magnitude and phase of the frequency response as the output. The Nyquist plot displays these in polar coordinates, with magnitude mapping to radius and phase to argument (angle). The Nichols plot displays these in rectangular coordinates, on the log scale.