Bode plot: Difference between revisions

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{{Short description|Graph of the frequency response of a control system}}

{{refimprove|date=December 2011}}

[[File:High-Pass filter Bode Magnitude and Phase plots.png|class=skin-invert-image|thumb|326x326px|Figure 1A: [[High-pass filter]] (1st-order, one-pole) Bode magnitude plot (top) and Bode phase plot (bottom). The red data curve is approximated by the straight black line.]]

[[File:Low-Pass filter Bode Magnitude and Phase plots.png|class=skin-invert-image|thumb|331x331px|Figure 1B: [[Low-pass filter]] (1st-order, one-pole) Bode magnitude plot (top) and Bode phase plot (bottom). The red data curve is approximated by the straight black line.]]

In [[electrical engineering]] and [[control theory]], a '''Bode plot''' {{IPAc-en|ˈ|b|oʊ|d|i|}} is a [[plot (graphics)|graph]] of the [[frequency response]] of a system. It is usually a combination of a '''Bode magnitude plot''', expressing the magnitude (usually in [[decibel]]s) of the frequency response, and a '''Bode phase plot''', expressing the [[phase (waves)|phase shift]].

As originally conceived by [[Hendrik Wade Bode]] in the 1930s, the plot is an [[asymptotic]] [[approximation]] of the frequency response, [[piecewise linear function|using straight line segments]].<ref name="Yarlagadda2010">{{cite book|author=R. K. Rao Yarlagadda|title=Analog and Digital Signals and Systems|url=https://archive.org/details/analogdigitalsig00yarl_849|url-access=limited|year=2010|publisher=Springer Science & Business Media|isbn=978-1-4419-0034-0|page=[https://archive.org/details/analogdigitalsig00yarl_849/page/n271 243]}}</ref>

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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 {{IPAc-en|ˈ|b|oʊ|d|i}} {{respell|BOH|dee}}, althoughwhereas thein Dutch pronunciationit is Bo-duh.usually ({{IPA-|nl|ˈboːdə|lang}}), closer to English {{IPAc-en|ˈ|b|oʊ|d|ə}} {{respell|BOH|də}}, which is preferred by his family, but less common among researchers.<ref name="Van Valkenburg">Van Valkenburg, M. E. University of Illinois at Urbana-Champaign, "In memoriam: Hendrik W. Bode (1905-1982)", [[IEEE]] Transactions on Automatic Control, Vol.&nbsp;AC-29, No&nbsp;3., March 1984, pp.&nbsp;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."</ref><ref>{{cite web |title=Vertaling van postbode, NL>EN |url= http://www.mijnwoordenboek.nl/vertalen.php?s1=&s2=NL+%3E+EN&s3=NL+%3E+EN&woord=postbode |publisher=mijnwoordenboek.nl |access-date=2013-10-07}}</ref><!--Should this be moved to the beginning of the page? I have no preference, but pronounciation information is usually listed at the beginning of an article.-->

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.&nbsp;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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The '''Bode magnitude plot''' is the graph of the function <math> |H(s=j \omega)|</math> of frequency <math> \omega</math> (with <math>j</math> being the [[imaginary unit]]). The <math> \omega</math>-axis of the magnitude plot is logarithmic and the magnitude is given in [[decibel]]s, i.e., a value for the magnitude <math>|H|</math> is plotted on the axis at <math>20 \log_{10} |H|</math>.

The '''Bode phase plot''' is the graph of the [[Argument (complex analysis)|phase]], commonly expressed in degrees, of the transfer[[Argument (complex analysis)|argument function]] <math> \arg \left( H(s =j \omega) \right)</math> as a function of <math>\omega</math>. The phase is plotted on the same logarithmic <math>\omega</math>-axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.

==Frequency response ==

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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 2-52–5 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&nbsp;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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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&nbsp;dB/decade drop of the pole is arrested by the 20&nbsp;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.

<gallery caption="Example with pole and zero" widths="300px" perrow="2" class="skin-invert-image">

Image:Bode Low Pass Magnitude Plot.PNG|Figure 2: Bode magnitude plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots

Image:Bode Low Pass Phase Plot.PNG|Figure 3: Bode phase plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots

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==Gain margin and phase margin==

{{See also|Phase margin}}

Bode plots are used to assess the stability of [[negative -feedback amplifier]]s by finding the gain and [[phase margin]]s of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by

:<math>A_\mathrmtext{FB} = \frac {A_\mathrmtext{OL}} {1 + \beta A_\mathrmtext{OL}} \; ,</math>

where ''A''_FB is the gain of the amplifier with feedback (the '''closed-loop gain'''), ''β'' is the '''feedback factor''', and ''A''_OL is the gain without feedback (the '''open-loop gain'''). The gain ''A''_OL 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''_OL = −1. (Thatthat is, the magnitude of β''A''_OL 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''₁₈₀, is the frequency where the [[open-loop gain]] flips sign. The second, labeled here ''f''_0&nbsp;dB, is the frequency where the magnitude of the product | β ''A''_OL | = 1 (in dB, magnitude 1 is= 0&nbsp;dB). That is, frequency ''f''₁₈₀ is determined by the condition:

:<math>\beta A_\mathrmtext{OL} \left( f_{180} \right) = - | \beta A_\mathrmtext{OL} \left( f_{180} \right) | = - | \beta A_\mathrmtext{OL}|_{180} \; ,</math>

where vertical bars denote the [[Absolute value#Complex numbers|magnitude of a complex number (for example, <math>|a + \mathrm{j}b| = \left[ a^2 + b^2 \right]^{\frac12}</math>)], and frequency ''f''_0&nbsp;dB is determined by the condition:

:<math>| \beta A_\mathrmtext{OL} \left( f_\mathrmtext{0dB0 dB} \right) | = 1 \; .</math>

One measure of proximity to instability is the '''gain margin'''. The Bode phase plot locates the frequency where the phase of β''A''_OL reaches &minus;180°, denoted here as frequency ''f''₁₈₀. Using this frequency, the Bode magnitude plot finds the magnitude of β''A''_OL. If |β''A''_OL|₁₈₀ ≥ 1, the amplifier is unstable, as mentioned. If |β''A''_OL|₁₈₀ < 1, instability does not occur, and the separation in dB of the magnitude of |β''A''_OL|₁₈₀ from |β''A''_OL| = 1 is called the ''gain margin''. Because a magnitude of one1 is 0&nbsp;dB, the gain margin is simply one of the equivalent forms: <math>20 \log_{10} ( | \beta A_\mathrmtext{OL} |_{180} ) = 20 \log_{10} ( |A_\mathrmtext{OL}|) - 20 \log_{10} (\beta^{-1})</math>.

Another equivalent measure of proximity to instability is the '''[[phase margin]]'''. The Bode magnitude plot locates the frequency where the magnitude of |β''A''_OL| reaches unity, denoted here as frequency ''f''_0&nbsp;dB. Using this frequency, the Bode phase plot finds the phase of β''A''_OL. If the phase of β''A''_OL( ''f''_0&nbsp;dB) > &minus;180°, the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when ''f'' = ''f''₁₈₀), and the distance of the phase at ''f''_0&nbsp;dB in degrees above &minus;180° is called the ''phase margin''.

If a simple ''yes'' or ''no'' on the stability issue is all that is needed, the amplifier is stable if ''f''_0&nbsp;dB < ''f''₁₈₀. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions ([[minimum phase]] systems). Although these restrictions usually are met, if they are not, then another method must be used, such as the [[Nyquist plot]].<ref name=Lee>

{{cite book

|author=Thomas H. Lee

|title=The design of CMOS radio-frequency integrated circuits

|section=§14.6. Gain and Phase Margins as Stability Measures

|pagepages=§14.6 pp. 451–453

|year= 2004

|edition=Second2nd

|publisher=Cambridge University Press

|___location=Cambridge UK

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</ref><ref name=Levine>

{{cite book

|author=William S. Levine

|title=The control handbook: the electrical engineering handbook series

|section=§10.1. Specifications of Control System

|page=§10.1 p. 163

|year= 1996

|edition=Second2nd

|publisher=CRC Press/IEEE Press

|___location=Boca Raton FL

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|author=Allen Tannenbaum | author-link = Allen Tannenbaum

|title=Invariance and Systems Theory: Algebraic and Geometric Aspects

| date = February 1981

|publisher=Springer-Verlag

|___location=New York, NY

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Stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good [[Step response#Step response of feedback amplifiers|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.<ref name=Sansen/> See also the discussion of phase margin in the [[Step response#Phase margin|step response]] article.

<gallery caption="Examples" widths="300px" perrow="2" class="skin-invert-image">

Image:Magnitude of feedback amplifier.PNG|Figure 6: Gain of feedback amplifier ''A''_FB in dB and corresponding open-loop amplifier ''A''_OL. Parameter 1/β = 58&nbsp;dB, and at low frequencies ''A''_FB ≈ 58&nbsp;dB as well. The gain margin in this amplifier is nearly zero because &#124; β''A''_OL&#124; = 1 occurs at almost ''f'' = ''f''_180°.

Image:Phase of feedback amplifier.PNG|Figure 7: Phase of feedback amplifier ''°A''_FB in degrees and corresponding open-loop amplifier ''°A''_OL. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency ''f'' = ''f''_0&nbsp;dB where &#124; β''A''_OL&#124; = 1.

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==Bode plotter==

[[Image:Bodeplot.png|class=skin-invert-image|380px|thumb|right|Figure 10: Amplitude diagram of a 10th -order [[electronic filter]] plotted using a Bode Plotter application.plotter]]

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 &nbsp;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).

== Related plots ==

{{main article|Nyquist plot|Nichols plot}}

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]].

<gallery mode="packed" heights="175" class="skin-invert-image">

Image:Nyquist plot.svg|Figure 11: A [[Nyquist plot]].

Image:Nichols plot.svg|Figure 12: A [[Nichols plot]] of the same response from Figure 11.

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* [[Bode's sensitivity integral]]

*[[Kramers–Kronig relations#Magnitude (gain)–phase relation|Bode's magnitude (gain)–phase relation]]

* [[Electrochemical impedanceDielectric spectroscopy]]

== Notes ==