Bode plot

In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system.

As originally conceived by Hendrik Wade Bode in the 1930s, the plot is an asymptotic approximation of the frequency response, using straight line segments.

[1] 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.

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.

[4] The principles developed were applied to design problems of servomechanisms and other feedback control systems.

This section illustrates that a Bode plot is a visualization of the frequency response of a system.

It can be shown[5] that the magnitude of the response is and that the phase shift is In summary, subjected to an input with frequency

For many practical problems, the detailed Bode plots can be approximated with straight-line segments that are asymptotes of the precise response.

The effect of each of the terms of a multiple element transfer function can be approximated by a set of straight lines on a Bode plot.

Before widespread availability of digital computers, graphical methods were extensively used to reduce the need for tedious calculation; a graphical solution could be used to identify feasible ranges of parameters for a new design.

The method for drawing amplitude plots implicitly uses this idea, but since the log of the amplitude of each pole or zero always starts at zero and only has one asymptote change (the straight lines), the method can be simplified.

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.

Given a transfer function in the same form as above, the idea is to draw separate plots for each pole and zero, then add them up.

The actual phase curve is given by To draw the phase plot, for each pole and zero: 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: The Bode plot is shown in Figure 1(b) above, and construction of the straight-line approximation is discussed next.

The straight-line plots are horizontal up to the pole (zero) location and then drop (rise) at 20 dB/decade.

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°.

The zero has been moved to higher frequency than the pole to make a more interesting example.

[note 1] Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product βAOL = −1 (that is, the magnitude of βAOL is unity and its phase is −180°, the so-called Barkhausen stability criterion).

That is, frequency f180 is determined by the condition where vertical bars denote the magnitude of a complex number, and frequency f0 dB is determined by the condition One measure of proximity to instability is the gain margin.

If |βAOL|180 < 1, instability does not occur, and the separation in dB of the magnitude of |βAOL|180 from |βAOL| = 1 is called the gain margin.

This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions (minimum phase systems).

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).

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 feedback factor is chosen smaller than in Figure 6 or 7, moving the condition | β AOL | = 1 to lower frequency.

Using Figure 9, for a phase of −180° the value of f180 = 3.332 kHz (the same result as found earlier, of course[note 3]).

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.

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.

For education and research purposes, plotting Bode diagrams for given transfer functions facilitates better understanding and getting faster results (see external links).

The Nyquist plot displays these in polar coordinates, with magnitude mapping to radius and phase to argument (angle).