In electrical engineering and control theory, a Bode plot 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.
Overview
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 in English as , whereas in Dutch it is usually , closer to English , which is preferred by his family, but less common among researchers.Van Valkenburg, M. E. University of Illinois at Urbana-Champaign, "In memoriam: Hendrik W. Bode (1905-1982)", IEEE Transactions on Automatic Control, Vol. AC-29, No 3., March 1984, pp. 193–194. Quote: "Something should be said about his name. To his colleagues at Bell Laboratories and the generations of engineers that have followed, the pronunciation is boh-dee. The Bode family preferred that the original Dutch be used as boh-dah."
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.David A. Mindell Between Human and Machine: Feedback, Control, and Computing Before Cybernetics JHU Press, 2004, , pp. 127–131. 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.
Definition 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 argument 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.
Frequency response This section illustrates that a Bode plot is a visualization of the frequency response of a system.
Consider a linear, time-invariant system with transfer function . Assume that the system is subject to a sinusoidal input with frequency ,
that is applied persistently, i.e. from a time to a time . The response will be of the form
i.e., also a sinusoidal signal with amplitude shifted by a phase with respect to the input.
It can be shown that the magnitude of the response is
and that the phase shift is
In summary, subjected to an input with frequency , the system responds at the same frequency with an output that is amplified by a factor and phase-shifted by . These quantities, thus, characterize the frequency response and are shown in the Bode plot.
Rules for handmade Bode plot
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. This allows a graphical solution of the overall frequency response function. 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 premise of a Bode plot is that one can consider the log of a function in the form
as a sum of the logs of its zeros and poles:
This idea is used explicitly in the method for drawing phase diagrams. 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.
Straight-line amplitude plot Amplitude decibels is usually done using to define decibels. Given a transfer function in the form
where and are constants, , , and is the transfer function: At every value of s where (a zero), increase the slope of the line by per decade. At every value of s where (a pole), decrease the slope of the line by per decade. The initial value of the graph depends on the boundaries. The initial point is found by putting the initial angular frequency into the function and finding The initial slope of the function at the initial value depends on the number and order of zeros and poles that are at values below the initial value, and is found using the first two rules.
To handle irreducible 2nd-order polynomials, can, in many cases, be approximated as .
Note that zeros and poles happen when is equal to a certain or . This is because the function in question is the magnitude of , and since it is a complex function, . Thus at any place where there is a zero or pole involving the term , the magnitude of that term is .
Corrected amplitude plot
To correct a straight-line amplitude plot: At every zero, put a point above the line. At every pole, put a point below the line. 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 or . 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
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: If is positive, start line (with zero slope) at 0°. If is negative, start line (with zero slope) at −180°. If the sum of the number of unstable zeros and poles is odd, add 180° to that basis. At every (for stable zeros ), increase the slope by degrees per decade, beginning one decade before (e.g., ). At every (for stable poles ), decrease the slope by degrees per decade, beginning one decade before (e.g., ). "Unstable" (right half-plane) poles and zeros () have opposite behavior. Flatten the slope again when the phase has changed by degrees (for a zero) or degrees (for a pole). After plotting one line for each pole or zero, add the lines together to obtain the final phase plot; that is, the final phase plot is the superposition of each earlier phase plot.
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:
The Bode plot is shown in Figure 1(b) above, and construction of the straight-line approximation is discussed next.
Magnitude plot The magnitude (in decibels) of the transfer function above (normalized and converted to angular-frequency form), given by the decibel gain expression :
Then plotted versus input frequency on a logarithmic scale, can be approximated by two lines, forming the asymptotic (approximate) magnitude Bode plot of the transfer function: The first line for angular frequencies below is a horizontal line at 0 dB, since at low frequencies the term is small and can be neglected, making the decibel gain equation above equal to zero. The second line for angular frequencies above is a line with a slope of −20 dB per decade, since at high frequencies the term dominates, and the decibel gain expression above simplifies to , which is a straight line with a slope of −20 dB per decade.
These two lines meet at the corner frequency . From the plot, it can be seen that for frequencies well below the corner frequency, the circuit has an attenuation of 0 dB, corresponding to a unity pass-band gain, i.e. the amplitude of the filter output equals the amplitude of the input. Frequencies above the corner frequency are attenuated the higher the frequency, the higher the attenuation.
Phase plot The phase Bode plot is obtained by plotting the phase angle of the transfer function given by
versus , where and are the input and cutoff angular frequencies respectively. For input frequencies much lower than corner, the ratio is small, and therefore the phase angle is close to zero. As the ratio increases, the absolute value of the phase increases and becomes −45° when . As the ratio increases for input frequencies much greater than the corner frequency, the phase angle asymptotically approaches −90°. The frequency scale for the phase plot is logarithmic.
Normalized plot 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 .
An example with zero and pole 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.
Gain margin and phase margin
Bode plots are used to assess the stability of negative-feedback amplifiers by finding the gain and phase margins of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by