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== Overview ==
Among his several important contributions to [[Network analysis (electrical circuits)|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 (electronics)|gain]] and phase-shift plots. These bear his name, ''Bode gain plot'' and ''Bode phase plot''. "Bode" is often pronounced in English as {{
Bode was faced with the problem of designing stable [[amplifier]]s 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.<ref>David A. Mindell ''Between Human and Machine: Feedback, Control, and Computing Before Cybernetics'' JHU Press, 2004, {{ISBN|0801880572}}, pp. 127–131.</ref> The principles developed were applied to design problems of [[servomechanism]]s and other feedback control systems. The Bode plot is an example of analysis in the [[frequency ___domain]].
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* Draw a smooth curve through those points using the straight lines as asymptotes (lines which the curve approaches).
Note that this correction method does not incorporate how to handle complex values of <math>x_n</math> or <math>y_n</math>. In the case of an [[irreducible polynomial]], the best way to correct the plot is to actually calculate the magnitude of the transfer function at the pole or zero corresponding to the irreducible polynomial, and put that dot over or under the line at that pole or zero.
=== Straight-line phase plot ===
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==Example==
To create a straight-line plot for a first-order (one-pole) low-pass filter, one considers the normalized form of the transfer function in terms of the [[angular frequency]]:
:<math>H_{\text{lp}}(\mathrm{j} \omega) = \frac{1}{1 + \mathrm{j} \frac{\omega}{\omega_\text{c}}}.</math>
The Bode plot is shown in Figure 1(b) above, and construction of the straight-line approximation is discussed next.
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==An example with zero and pole ==
Figures
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°.
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where ''A''<sub>FB</sub> is the gain of the amplifier with feedback (the ''closed-loop gain''), ''β'' is the ''feedback factor'', and ''A''<sub>OL</sub> is the gain without feedback (the ''open-loop gain''). The gain ''A''<sub>OL</sub> is a complex function of frequency, with both magnitude and phase.<ref group="note">Ordinarily, as frequency increases, the magnitude of the gain drops, and the phase becomes more negative, although these are only trends and may be reversed in particular frequency ranges. Unusual gain behavior can render the concepts of gain and phase margin inapplicable. Then other methods such as the [[Nyquist plot]] have to be used to assess stability.</ref> Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product β''A''<sub>OL</sub> = −1 (that is, the magnitude of β''A''<sub>OL</sub> is unity and its phase is −180°, the so-called [[Barkhausen stability criterion]]). Bode plots are used to determine just how close an amplifier comes to satisfying this condition.
Key to this determination are two frequencies. The first, labeled here as ''f''<sub>180</sub>, is the frequency where the [[open-loop gain]] flips sign. The second, labeled here ''f''<sub>0 dB</sub>, is the frequency where the magnitude of the product |β''A''<sub>OL</sub>| = 1 = 0 dB. That is, frequency ''f''<sub>180</sub> is determined by the condition
:<math>\beta A_\text{OL}(f_{180}) = -|\beta A_\text{OL}(f_{180})| = -|\beta A_\text{OL}|_{180},</math>
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