4488:
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.
5576:
5627:
4522:
4534:
4498:
4510:
5671:
133:
5697:
145:
3916:
5825:
5809:
36:
3642:
4163:. 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
3911:{\displaystyle {\begin{aligned}A_{\text{vdB}}&=20\log |H_{\text{lp}}(\mathrm {j} \omega )|\\&=20\log {\frac {1}{\left|1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}\right|}}\\&=-20\log \left|1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}\right|\\&=-10\log \left(1+{\frac {\omega ^{2}}{\omega _{\text{c}}^{2}}}\right).\end{aligned}}}
5575:
5626:
5884:
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
1512:
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.
5990:
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
4483:
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
1312:
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
4487:
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
4484:
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°.
4521:
4533:
2678:. 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.
3592:
2518:
5139:
4854:
4262:
5670:
5562:, 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
2914:
4632:
2808:
1693:
5696:
4497:
2369:
5416:
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,
4509:
1507:
4933:
1174:
4127:
3647:
989:
4402:. 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.
711:
2187:
2090:
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.
1104:
1391:
3267:
3058:
3513:
3430:
1569:
1294:
4443:
4400:
4367:
4074:
4009:
5472:
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
4445:. 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
4480:
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.
2084:
1254:
5933:
also does not change with a change in β. Therefore, we could use the previous values from
Figures 6 and 7. However, for clarity the procedure is described using only Figures 8 and 9.
2413:
373:
required to maintain stability under variations in circuit characteristics caused during manufacture or during operation. The principles developed were applied to design problems of
4527:
Figure 4: Bode magnitude plot for pole-zero combination; the location of the zero is ten times higher than in
Figures 2 and 3; curves labeled "Bode" are the straight-line Bode plots
2302:
4674:
is a complex function of frequency, with both magnitude and phase. Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product β
3498:
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.
1781:
2016:
1930:
871:
2612:
2568:
5010:
4539:
Figure 5: Bode phase plot for pole-zero combination; the location of the zero is ten times higher than in
Figures 2 and 3; curves labeled "Bode" are the straight-line Bode plots
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4332:
4161:
4039:
3974:
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1827:
5727:
4181:
3344:
3226:
3135:
3017:
646:
502:
3634:
1974:
1888:
5760:
For education and research purposes, plotting Bode diagrams for given transfer functions facilitates better understanding and getting faster results (see external links).
3507:
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:
2137:
3389:
3180:
2408:
5746:
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
1039:
897:
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795:
423:
600:
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5252:
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
5371:, 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
4569:
4369:
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
2692:
1577:
152:(1st-order, one-pole) Bode magnitude plot (top) and Bode phase plot (bottom). The red data curve is approximated by the straight black line.
140:(1st-order, one-pole) Bode magnitude plot (top) and Bode phase plot (bottom). The red data curve is approximated by the straight black line.
5860:
4410:
The horizontal frequency axis, in both the magnitude and phase plots, can be replaced by the normalized (nondimensional) frequency ratio
5232:. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions (
6096:
6001:
5824:
5798:, with magnitude mapping to radius and phase to argument (angle). The Nichols plot displays these in rectangular coordinates, on the
2307:
5554:
Stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good
5794:, with frequency as the input and magnitude and phase of the frequency response as the output. The Nyquist plot displays these in
4563:
of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by
1406:
5865:
5185:) > −180°, the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when
4876:
253:
2021:
The initial value of the graph depends on the boundaries. The initial point is found by putting the initial angular frequency
5971:
3609:) of the transfer function above (normalized and converted to angular-frequency form), given by the decibel gain expression
5409:
Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency
1130:
4079:
925:
100:
6239:
5236:
systems). Although these restrictions usually are met, if they are not, then another method must be used, such as the
72:
6234:
6164:
6134:
6106:
6079:
6052:
5646:. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency
5241:
662:
346:
306:
194:
119:
5469:≈ 3.332 kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.
3597:
The Bode plot is shown in Figure 1(b) above, and construction of the straight-line approximation is discussed next.
3587:{\displaystyle H_{\text{lp}}(\mathrm {j} \omega )={\frac {1}{1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}}}.}
2142:
1052:
5926:
does not change with a change in feedback factor; it is a property of the open-loop gain. The value of the gain at
4334:
are the input and cutoff angular frequencies respectively. For input frequencies much lower than corner, the ratio
2620:
Draw a smooth curve through those points using the straight lines as asymptotes (lines which the curve approaches).
1322:
5555:
3231:
3022:
3397:
1524:
1259:
79:
4503:
Figure 2: Bode magnitude plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots
4413:
4372:
4337:
4044:
3979:
2813:
the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by
2513:{\displaystyle {\sqrt {(x_{n}+\mathrm {j} \omega )(x_{n}-\mathrm {j} \omega )}}={\sqrt {x_{n}^{2}+\omega ^{2}}}}
6229:
6219:
6030:
5855:
5340:≈ 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain
4689:
57:
5986:
Van
Valkenburg, M. E. University of Illinois at Urbana-Champaign, "In memoriam: Hendrik W. Bode (1905-1982)",
2045:
1304:
For many practical problems, the detailed Bode plots can be approximated with straight-line segments that are
1215:
6224:
4515:
Figure 3: Bode phase plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots
2273:
6214:
5134:{\displaystyle 20\log _{10}|\beta A_{\text{OL}}|_{180}=20\log _{10}|A_{\text{OL}}|-20\log _{10}\beta ^{-1}}
1755:
86:
4849:{\displaystyle \beta A_{\text{OL}}(f_{180})=-|\beta A_{\text{OL}}(f_{180})|=-|\beta A_{\text{OL}}|_{180},}
1983:
1897:
823:
6189:
5275:
4556:
2579:
2535:
656:
365:
for use in telephone networks. He developed the graphical design technique of the Bode plots to show the
5907:
5742:, which produces a Bode diagram, or a graph, of a circuit's voltage gain or phase shift plotted against
4448:
4310:
4257:{\displaystyle \arg H_{\text{lp}}(\mathrm {j} \omega )=-\tan ^{-1}{\frac {\omega }{\omega _{\text{c}}}}}
4139:
4017:
3952:
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53:
17:
1786:
68:
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241:
5750:
control systems, through the measurement of corner (cutoff) frequencies and gain and phase margins.
5544:= 3.332 kHz (the same result as found earlier, of course). The open-loop gain from Figure 8 at
4692:). Bode plots are used to determine just how close an amplifier comes to satisfying this condition.
3612:
761:
This section illustrates that a Bode plot is a visualization of the frequency response of a system.
5963:
5845:
1946:
1860:
5808:
132:
5754:
2097:
144:
46:
5441:
Comparing the labeled points in Figure 6 and Figure 7, it is seen that the unity gain frequency
5007:. Because a magnitude of 1 is 0 dB, the gain margin is simply one of the equivalent forms:
3349:
3140:
4860:
2374:
753:-axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.
157:
4175:
The phase Bode plot is obtained by plotting the phase angle of the transfer function given by
1296:. These quantities, thus, characterize the frequency response and are shown in the Bode plot.
1024:
879:
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4270:
3924:
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3438:
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2195:
2024:
1195:
800:
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551:
507:
5955:
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1701:
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5563:
771:
399:
8:
5956:
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term is small and can be neglected, making the decibel gain equation above equal to zero.
1212:, the system responds at the same frequency with an output that is amplified by a factor
575:
2909:{\displaystyle \varphi (s)=-\arctan {\frac {\operatorname {Im} }{\operatorname {Re} }}.}
5779:
5690:. In this example, 1 / β = 77 dB. The gain margin in this amplifier is 19 dB.
5602:≈ 58 dB as well. The gain margin in this amplifier is nearly zero because | β
2953:
2930:
1933:
1832:
902:
527:
428:
269:
207:
93:
5479:| = 1 to lower frequency. In this example, 1 / β = 77 dB, and at low frequencies
2624:
Note that this correction method does not incorporate how to handle complex values of
1316:
The premise of a Bode plot is that one can consider the log of a function in the form
6160:
6130:
6102:
6075:
6048:
6026:
5967:
5795:
5731:
5527:= 1 kHz found above from the magnitude plot of Figure 8, the open-loop phase at
3945:, forming the asymptotic (approximate) magnitude Bode plot of the transfer function:
1309:
765:
394:
390:
261:
230:
2973:
If the sum of the number of unstable zeros and poles is odd, add 180° to that basis.
6122:
5791:
4985:< 1, instability does not occur, and the separation in dB of the magnitude of |β
4134:
378:
377:
and other feedback control systems. The Bode plot is an example of analysis in the
323:
283:
171:
137:
4702:, is the frequency where the open-loop gain flips sign. The second, labeled here
1397:
203:
149:
4041:
is a line with a slope of −20 dB per decade, since at high frequencies the
6183:
1308:
of the precise response. The effect of each of the terms of a multiple element
545:
446:
374:
316:
257:
161:
6023:
Between Human and
Machine: Feedback, Control, and Computing Before Cybernetics
5288:= 100 dB at low frequencies, and 1 / β = 58 dB. At low frequencies,
6198:
6068:
Thomas H. Lee (2004). "§14.6. Gain and Phase
Margins as Stability Measures".
5559:
5354:
is to look where the feedback gain intersects the open-loop gain. (Frequency
5233:
4627:{\displaystyle A_{\text{FB}}={\frac {A_{\text{OL}}}{1+\beta A_{\text{OL}}}},}
237:
223:
6074:(2nd ed.). Cambridge UK: Cambridge University Press. pp. 451–453.
5399:(examine the formula at the beginning of this section for the case of small
5886:
5850:
5831:
5815:
5787:
5783:
5773:
5769:
5739:
5489:
Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and
5237:
5146:
4560:
4550:
2803:{\displaystyle H(s)=A\prod {\frac {(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}},}
1688:{\displaystyle H(s)=A\prod {\frac {(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}},}
370:
4695:
Key to this determination are two frequencies. The first, labeled here as
1313:
could be used to identify feasible ranges of parameters for a new design.
6154:
6069:
5757:, but the network analyzer is typically used at much higher frequencies.
5551:
is 58 dB, and 1 / β = 77 dB, so the gain margin is 19 dB.
5218:
on the stability issue is all that is needed, the amplifier is stable if
5150:. The Bode magnitude plot locates the frequency where the magnitude of |β
4164:
797:. Assume that the system is subject to a sinusoidal input with frequency
568:-axis of the magnitude plot is logarithmic and the magnitude is given in
366:
4956:. Using this frequency, the Bode magnitude plot finds the magnitude of β
234:
6095:
William S. Levine (1996). "§10.1. Specifications of
Control System".
5799:
5743:
2371:. Thus at any place where there is a zero or pole involving the term
1305:
358:
265:
4076:
term dominates, and the decibel gain expression above simplifies to
35:
5747:
362:
6101:(2nd ed.). Boca Raton FL: CRC Press/IEEE Press. p. 163.
4129:, which is a straight line with a slope of −20 dB per decade.
3606:
569:
215:
6098:
The control handbook: the electrical engineering handbook series
6047:. Chichester, West Sussex, England: John Wiley & Sons, Ltd.
5164:. Using this frequency, the Bode phase plot finds the phase of β
4942:. The Bode phase plot locates the frequency where the phase of β
3976:
is a horizontal line at 0 dB, since at low frequencies the
268:
in the 1930s, devised a simple but accurate method for graphing
2364:{\displaystyle |H(\mathrm {j} \omega )|={\sqrt {H\cdot H^{*}}}}
2270:. This is because the function in question is the magnitude of
6127:
Invariance and
Systems Theory: Algebraic and Geometric Aspects
5144:
Another equivalent measure of proximity to instability is the
5987:
5898:
The critical amount of feedback where the peak in the gain
5738:
The Bode plotter is an electronic instrument resembling an
5726:
1502:{\displaystyle \log(f(x))=\log(A)+\sum a_{n}\log(x-c_{n}).}
338:
332:
298:
292:
186:
180:
6159:. Dordrecht, The Netherlands: Springer. pp. 157–163.
5778:
Two related plots that display the same data in different
1571:
to define decibels. Given a transfer function in the form
659:, commonly expressed in degrees, of the transfer function
4928:{\displaystyle |\beta A_{\text{OL}}(f_{\text{0 dB}})|=1.}
4709:, is the frequency where the magnitude of the product |β
5240:. Optimal gain and phase margins may be computed using
6071:
The design of CMOS radio-frequency integrated circuits
5534:
is −135°, which is a phase margin of 45° above −180°.
3435:
Flatten the slope again when the phase has changed by
1044:
It can be shown that the magnitude of the response is
5595:. Parameter 1/β = 58 dB, and at low frequencies
5326:. The feedback gain at low frequencies and for large
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Given a transfer function in the same form as above,
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1169:{\displaystyle \varphi =\arg H(\mathrm {j} \omega ).}
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Bode was faced with the problem of designing stable
335:
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183:
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4974:≥ 1, the amplifier is unstable, as mentioned. If |β
4122:{\displaystyle -20\log(\omega /\omega _{\text{c}})}
2970:
is negative, start line (with zero slope) at −180°.
326:
286:
174:
60:. Unsourced material may be challenged and removed.
27:
Graph of the frequency response of a control system
6036:
5919:The frequency where the open-loop gain flips sign
5537:Using Figure 9, for a phase of −180° the value of
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984:{\displaystyle y(t)=y_{0}\sin(\omega t+\varphi ),}
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5962:. Springer Science & Business Media. p.
5753:This is identical to the function performed by a
5709:in degrees and corresponding open-loop amplifier
5639:in degrees and corresponding open-loop amplifier
1192:In summary, subjected to an input with frequency
449:) consists of a magnitude plot and a phase plot.
6196:
6152:
6121:
5385:, the open-loop gain is sufficiently small that
4644:is the gain of the amplifier with feedback (the
4475:
3303:degrees per decade, beginning one decade before
3094:degrees per decade, beginning one decade before
2947:is positive, start line (with zero slope) at 0°.
5520:Figure 9 is the phase plot. Using the value of
5503:= 1 kHz. Notice that the peak in the gain
4949:reaches −180°, denoted here as frequency
4938:One measure of proximity to instability is the
4688:is unity and its phase is −180°, the so-called
4555:Bode plots are used to assess the stability of
4544:
3941:on a logarithmic scale, can be approximated by
3394:"Unstable" (right half-plane) poles and zeros (
1516:
1299:
876:that is applied persistently, i.e. from a time
733:. The phase is plotted on the same logarithmic
706:{\displaystyle \arg \left(H(s=j\omega )\right)}
210:of a system. It is usually a combination of a
4014:The second line for angular frequencies above
2182:{\displaystyle ({\sqrt {a}}x+{\sqrt {c}})^{2}}
1099:{\displaystyle y_{0}=|H(\mathrm {j} \omega )|}
994:i.e., also a sinusoidal signal with amplitude
6067:
5980:
5947:
5730:Figure 10: Amplitude diagram of a 10th-order
3949:The first line for angular frequencies below
2094:To handle irreducible 2nd-order polynomials,
1386:{\displaystyle f(x)=A\prod (x-c_{n})^{a_{n}}}
252:Among his several important contributions to
6043:Skogestad, Sigurd; Postlewaite, Ian (2005).
5716:. The phase margin in this amplifier is 45°.
5683:in dB and corresponding open-loop amplifier
5588:in dB and corresponding open-loop amplifier
5247:
3262:{\displaystyle -\operatorname {Re} (p)<0}
3053:{\displaystyle -\operatorname {Re} (z)<0}
272:and phase-shift plots. These bear his name,
6184:How to draw piecewise asymptotic Bode plots
5361:is needed later to find the phase margin.)
5203:in degrees above −180° is called the
5157:| reaches unity, denoted here as frequency
3425:{\displaystyle \operatorname {Re} (s)>0}
2681:
2528:To correct a straight-line amplitude plot:
2523:
1564:{\displaystyle {\text{dB}}=20\log _{10}(X)}
1289:{\displaystyle \arg H(\mathrm {j} \omega )}
4438:{\displaystyle \omega /\omega _{\text{c}}}
4395:{\displaystyle \omega =\omega _{\text{c}}}
4362:{\displaystyle \omega /\omega _{\text{c}}}
4069:{\displaystyle \omega /\omega _{\text{c}}}
4004:{\displaystyle \omega /\omega _{\text{c}}}
6148:
6146:
6115:
5455:are very nearly equal in this amplifier,
1521:Amplitude decibels is usually done using
120:Learn how and when to remove this message
5725:
5364:Near this crossover of the two gains at
2079:{\displaystyle |H(\mathrm {j} \omega )|}
1249:{\displaystyle |H(\mathrm {j} \omega )|}
143:
131:
6188:Gnuplot code for generating Bode plot:
5378:= −1). Beyond the unity gain frequency
2139:can, in many cases, be approximated as
214:, expressing the magnitude (usually in
14:
6197:
6143:
5958:Analog and Digital Signals and Systems
5866:Electrochemical impedance spectroscopy
5861:Bode's magnitude (gain)–phase relation
5702:Figure 9: Phase of feedback amplifier
5632:Figure 7: Phase of feedback amplifier
4716:| = 1 = 0 dB. That is, frequency
2304:, and since it is a complex function,
2297:{\displaystyle H(\mathrm {j} \omega )}
2192:Note that zeros and poles happen when
314:, although the Dutch pronunciation is
6088:
5676:Figure 8: Gain of feedback amplifier
5581:Figure 6: Gain of feedback amplifier
1776:{\displaystyle s=\mathrm {j} \omega }
756:
315:
5889:have to be used to assess stability.
5834:of the same response from Figure 11.
5196:), and the distance of the phase at
3921:Then plotted versus input frequency
2011:{\displaystyle 20b_{n}\ {\text{dB}}}
1925:{\displaystyle 20a_{n}\ {\text{dB}}}
1124:
1046:
866:{\displaystyle u(t)=\sin(\omega t),}
58:adding citations to reliable sources
29:
5991:original Dutch be used as boh-dah."
2607:{\displaystyle 3b_{n}\ {\text{dB}}}
2563:{\displaystyle 3a_{n}\ {\text{dB}}}
919:. The response will be of the form
445:being the complex frequency in the
218:) of the frequency response, and a
24:
6061:
6002:"Vertaling van postbode, NL>EN"
4663:is the gain without feedback (the
4465:{\displaystyle \omega _{\text{c}}}
4405:
4327:{\displaystyle \omega _{\text{c}}}
4205:
4156:{\displaystyle \omega _{\text{c}}}
4034:{\displaystyle \omega _{\text{c}}}
3969:{\displaystyle \omega _{\text{c}}}
3808:
3746:
3695:
3557:
3531:
2463:
2436:
2323:
2284:
2061:
1766:
1276:
1231:
1153:
1081:
886:
572:, i.e., a value for the magnitude
25:
6251:
6177:
6129:. New York, NY: Springer-Verlag.
5305:is plotted and not the product β
4681:= −1 (that is, the magnitude of β
3600:
5823:
5807:
5763:
5695:
5669:
5625:
5574:
5347:), so an equivalent way to find
4532:
4520:
4508:
4496:
2410:, the magnitude of that term is
1822:{\displaystyle a_{n},b_{n}>0}
322:
282:
170:
34:
5913:
5892:
5721:
4870:is determined by the condition
4859:where vertical bars denote the
4723:is determined by the condition
3339:{\displaystyle \omega =|y_{n}|}
3221:{\displaystyle \omega =|y_{n}|}
3130:{\displaystyle \omega =|x_{n}|}
3012:{\displaystyle \omega =|x_{n}|}
641:{\displaystyle 20\log _{10}|H|}
497:{\displaystyle |H(s=j\omega )|}
45:needs additional citations for
6190:DIN-A4 printing template (pdf)
6045:Multivariable Feedback Control
6015:
5994:
5878:
5095:
5080:
5050:
5031:
4915:
4911:
4898:
4881:
4833:
4814:
4803:
4799:
4786:
4769:
4759:
4746:
4690:Barkhausen stability criterion
4212:
4201:
4116:
4095:
3706:
3702:
3691:
3677:
3629:{\displaystyle A_{\text{vdB}}}
3538:
3527:
3413:
3407:
3369:
3354:
3332:
3317:
3250:
3244:
3214:
3199:
3160:
3145:
3123:
3108:
3041:
3035:
3005:
2990:
2897:
2894:
2888:
2882:
2871:
2868:
2862:
2856:
2832:
2826:
2778:
2758:
2740:
2720:
2705:
2699:
2470:
2446:
2443:
2419:
2397:
2378:
2334:
2330:
2319:
2312:
2291:
2280:
2170:
2146:
2072:
2068:
2057:
2050:
2041:into the function and finding
1663:
1643:
1625:
1605:
1590:
1584:
1558:
1552:
1493:
1474:
1449:
1443:
1431:
1428:
1422:
1416:
1367:
1347:
1335:
1329:
1283:
1272:
1242:
1238:
1227:
1220:
1160:
1149:
1092:
1088:
1077:
1070:
975:
960:
938:
932:
857:
848:
836:
830:
784:
778:
768:system with transfer function
695:
680:
634:
626:
588:
580:
490:
486:
471:
464:
412:
406:
13:
1:
5954:R. K. Rao Yarlagadda (2010).
5940:
5902:disappears altogether is the
5448:and the phase-flip frequency
5242:Nevanlinna–Pick interpolation
4861:magnitude of a complex number
4476:An example with zero and pole
4170:
1969:{\displaystyle \omega =y_{n}}
1883:{\displaystyle \omega =x_{n}}
456:is the graph of the function
384:
280:. "Bode" is often pronounced
233:in the 1930s, the plot is an
5734:plotted using a Bode plotter
5609:| = 1 occurs at almost
5263:with the gain with feedback
4557:negative-feedback amplifiers
4545:Gain margin and phase margin
4133:These two lines meet at the
2919:To draw the phase plot, for
1517:Straight-line amplitude plot
1396:as a sum of the logs of its
1300:Rules for handmade Bode plot
1122:and that the phase shift is
242:using straight line segments
7:
5856:Bode's sensitivity integral
5839:
5298:Because the open-loop gain
5276:negative feedback amplifier
2576:At every pole, put a point
2532:At every zero, put a point
2132:{\displaystyle ax^{2}+bx+c}
1182:
1112:
1041:with respect to the input.
247:
240:of the frequency response,
229:As originally conceived by
10:
6256:
5767:
4548:
4492:Example with pole and zero
3502:
3384:{\displaystyle |y_{n}|/10}
3175:{\displaystyle |x_{n}|/10}
1849:is the transfer function:
602:is plotted on the axis at
6240:Filter frequency response
6153:Willy M C Sansen (2006).
5248:Examples using Bode plots
3432:) have opposite behavior.
2403:{\displaystyle (s+x_{n})}
1980:the slope of the line by
1894:the slope of the line by
6235:Classical control theory
6156:Analog design essentials
5871:
5846:Analog signal processing
4559:by finding the gain and
3465:degrees (for a zero) or
2682:Straight-line phase plot
2524:Corrected amplitude plot
1034:{\displaystyle \varphi }
892:{\displaystyle -\infty }
5755:vector network analyzer
4300:{\displaystyle \omega }
4280:{\displaystyle \omega }
3934:{\displaystyle \omega }
3488:{\displaystyle 90b_{n}}
3458:{\displaystyle 90a_{n}}
3296:{\displaystyle 45b_{n}}
3087:{\displaystyle 45a_{n}}
2205:{\displaystyle \omega }
2034:{\displaystyle \omega }
1205:{\displaystyle \omega }
810:{\displaystyle \omega }
746:{\displaystyle \omega }
726:{\displaystyle \omega }
561:{\displaystyle \omega }
517:{\displaystyle \omega }
5735:
5486:≈ 77 dB as well.
5295:≈ 58 dB as well.
5135:
4929:
4850:
4628:
4466:
4439:
4396:
4363:
4328:
4301:
4281:
4258:
4157:
4123:
4070:
4035:
4005:
3970:
3935:
3912:
3630:
3588:
3489:
3459:
3426:
3385:
3340:
3297:
3263:
3222:
3176:
3131:
3088:
3054:
3013:
2964:
2941:
2910:
2804:
2672:
2645:
2608:
2564:
2514:
2404:
2365:
2298:
2264:
2237:
2206:
2183:
2133:
2080:
2035:
2012:
1970:
1926:
1884:
1843:
1823:
1777:
1746:
1719:
1689:
1565:
1503:
1387:
1290:
1250:
1206:
1170:
1100:
1035:
1015:
985:
913:
893:
867:
811:
791:
766:linear, time-invariant
747:
727:
707:
642:
596:
562:
538:
518:
498:
439:
419:
391:linear, time-invariant
158:electrical engineering
153:
141:
6230:Electrical parameters
6220:Electronic amplifiers
5729:
5136:
4930:
4851:
4629:
4467:
4440:
4397:
4364:
4329:
4302:
4282:
4259:
4158:
4124:
4071:
4036:
4006:
3971:
3936:
3913:
3631:
3589:
3495:degrees (for a pole).
3490:
3460:
3427:
3386:
3341:
3298:
3264:
3223:
3177:
3132:
3089:
3055:
3014:
2965:
2942:
2911:
2805:
2673:
2671:{\displaystyle y_{n}}
2646:
2644:{\displaystyle x_{n}}
2609:
2565:
2515:
2405:
2366:
2299:
2265:
2263:{\displaystyle y_{n}}
2238:
2236:{\displaystyle x_{n}}
2207:
2184:
2134:
2081:
2036:
2013:
1971:
1927:
1885:
1844:
1824:
1778:
1747:
1745:{\displaystyle y_{n}}
1720:
1718:{\displaystyle x_{n}}
1690:
1566:
1504:
1388:
1291:
1256:and phase-shifted by
1251:
1207:
1171:
1101:
1036:
1016:
1014:{\displaystyle y_{0}}
986:
914:
894:
868:
812:
792:
748:
728:
708:
643:
597:
563:
539:
519:
499:
440:
420:
147:
135:
6225:Electronics concepts
6033:, pp. 127–131.
6004:. mijnwoordenboek.nl
5011:
5003:| = 1 is called the
4877:
4730:
4570:
4449:
4414:
4373:
4338:
4311:
4291:
4271:
4182:
4140:
4080:
4045:
4018:
3980:
3953:
3925:
3643:
3613:
3514:
3469:
3439:
3398:
3350:
3307:
3277:
3232:
3189:
3141:
3098:
3068:
3023:
2980:
2954:
2931:
2820:
2693:
2655:
2628:
2580:
2536:
2414:
2375:
2308:
2274:
2247:
2220:
2196:
2143:
2098:
2046:
2025:
1984:
1947:
1898:
1861:
1833:
1787:
1756:
1729:
1702:
1578:
1525:
1407:
1323:
1260:
1216:
1196:
1131:
1053:
1025:
998:
926:
903:
880:
824:
801:
790:{\displaystyle H(s)}
772:
737:
717:
663:
655:is the graph of the
606:
576:
552:
528:
508:
460:
429:
418:{\displaystyle H(s)}
400:
389:The Bode plot for a
320:, closer to English
54:improve this article
6215:Electronic feedback
5171:. If the phase of β
3893:
2494:
1021:shifted by a phase
595:{\displaystyle |H|}
454:Bode magnitude plot
264:, while working at
212:Bode magnitude plot
5780:coordinate systems
5736:
5131:
4925:
4846:
4624:
4462:
4435:
4392:
4359:
4324:
4297:
4277:
4254:
4153:
4119:
4066:
4031:
4001:
3966:
3931:
3908:
3906:
3879:
3626:
3605:The magnitude (in
3584:
3485:
3455:
3422:
3381:
3336:
3293:
3259:
3228:(for stable poles
3218:
3172:
3127:
3084:
3050:
3019:(for stable zeros
3009:
2960:
2937:
2906:
2800:
2668:
2641:
2604:
2560:
2510:
2480:
2400:
2361:
2294:
2260:
2233:
2202:
2179:
2129:
2076:
2031:
2008:
1966:
1939:At every value of
1922:
1880:
1853:At every value of
1839:
1819:
1773:
1742:
1715:
1685:
1561:
1499:
1383:
1286:
1246:
1202:
1166:
1096:
1031:
1011:
981:
909:
889:
863:
807:
787:
757:Frequency response
743:
723:
703:
638:
592:
558:
534:
514:
494:
435:
415:
208:frequency response
154:
142:
6210:Signal processing
6125:(February 1981).
6025:JHU Press, 2004,
6021:David A. Mindell
5973:978-1-4419-0034-0
5796:polar coordinates
5732:electronic filter
5281:In this example,
5278:for more detail.
5091:
5045:
4908:
4895:
4828:
4783:
4743:
4619:
4615:
4595:
4580:
4459:
4432:
4389:
4356:
4321:
4252:
4249:
4198:
4150:
4113:
4063:
4028:
3998:
3963:
3894:
3886:
3827:
3824:
3772:
3765:
3762:
3688:
3657:
3623:
3579:
3576:
3573:
3524:
2963:{\displaystyle A}
2940:{\displaystyle A}
2901:
2795:
2602:
2598:
2558:
2554:
2508:
2473:
2359:
2167:
2154:
2006:
2002:
1920:
1916:
1842:{\displaystyle H}
1680:
1531:
1310:transfer function
1190:
1189:
1120:
1119:
912:{\displaystyle t}
713:as a function of
537:{\displaystyle j}
438:{\displaystyle s}
395:transfer function
262:Hendrik Wade Bode
231:Hendrik Wade Bode
222:, expressing the
130:
129:
122:
104:
16:(Redirected from
6247:
6205:Plots (graphics)
6171:
6170:
6150:
6141:
6140:
6123:Allen Tannenbaum
6119:
6113:
6112:
6092:
6086:
6085:
6065:
6059:
6058:
6040:
6034:
6019:
6013:
6012:
6010:
6009:
5998:
5992:
5984:
5978:
5977:
5961:
5951:
5934:
5917:
5911:
5896:
5890:
5882:
5827:
5811:
5792:parametric plots
5699:
5673:
5629:
5578:
5517:is almost gone.
5319:= 1 / β decides
5312:, the condition
5140:
5138:
5137:
5132:
5130:
5129:
5114:
5113:
5098:
5093:
5092:
5089:
5083:
5075:
5074:
5059:
5058:
5053:
5047:
5046:
5043:
5034:
5026:
5025:
4934:
4932:
4931:
4926:
4918:
4910:
4909:
4906:
4897:
4896:
4893:
4884:
4863:, and frequency
4855:
4853:
4852:
4847:
4842:
4841:
4836:
4830:
4829:
4826:
4817:
4806:
4798:
4797:
4785:
4784:
4781:
4772:
4758:
4757:
4745:
4744:
4741:
4646:closed-loop gain
4633:
4631:
4630:
4625:
4620:
4618:
4617:
4616:
4613:
4597:
4596:
4593:
4587:
4582:
4581:
4578:
4536:
4524:
4512:
4500:
4471:
4469:
4468:
4463:
4461:
4460:
4457:
4444:
4442:
4441:
4436:
4434:
4433:
4430:
4424:
4401:
4399:
4398:
4393:
4391:
4390:
4387:
4368:
4366:
4365:
4360:
4358:
4357:
4354:
4348:
4333:
4331:
4330:
4325:
4323:
4322:
4319:
4306:
4304:
4303:
4298:
4286:
4284:
4283:
4278:
4263:
4261:
4260:
4255:
4253:
4251:
4250:
4247:
4238:
4233:
4232:
4208:
4200:
4199:
4196:
4162:
4160:
4159:
4154:
4152:
4151:
4148:
4135:corner frequency
4128:
4126:
4125:
4120:
4115:
4114:
4111:
4105:
4075:
4073:
4072:
4067:
4065:
4064:
4061:
4055:
4040:
4038:
4037:
4032:
4030:
4029:
4026:
4010:
4008:
4007:
4002:
4000:
3999:
3996:
3990:
3975:
3973:
3972:
3967:
3965:
3964:
3961:
3940:
3938:
3937:
3932:
3917:
3915:
3914:
3909:
3907:
3900:
3896:
3895:
3892:
3887:
3884:
3878:
3877:
3868:
3837:
3833:
3829:
3828:
3826:
3825:
3822:
3813:
3811:
3777:
3773:
3771:
3767:
3766:
3764:
3763:
3760:
3751:
3749:
3730:
3713:
3709:
3698:
3690:
3689:
3686:
3680:
3659:
3658:
3655:
3635:
3633:
3632:
3627:
3625:
3624:
3621:
3593:
3591:
3590:
3585:
3580:
3578:
3577:
3575:
3574:
3571:
3562:
3560:
3545:
3534:
3526:
3525:
3522:
3494:
3492:
3491:
3486:
3484:
3483:
3464:
3462:
3461:
3456:
3454:
3453:
3431:
3429:
3428:
3423:
3390:
3388:
3387:
3382:
3377:
3372:
3367:
3366:
3357:
3345:
3343:
3342:
3337:
3335:
3330:
3329:
3320:
3302:
3300:
3299:
3294:
3292:
3291:
3268:
3266:
3265:
3260:
3227:
3225:
3224:
3219:
3217:
3212:
3211:
3202:
3181:
3179:
3178:
3173:
3168:
3163:
3158:
3157:
3148:
3136:
3134:
3133:
3128:
3126:
3121:
3120:
3111:
3093:
3091:
3090:
3085:
3083:
3082:
3059:
3057:
3056:
3051:
3018:
3016:
3015:
3010:
3008:
3003:
3002:
2993:
2969:
2967:
2966:
2961:
2946:
2944:
2943:
2938:
2915:
2913:
2912:
2907:
2902:
2900:
2874:
2848:
2809:
2807:
2806:
2801:
2796:
2794:
2793:
2792:
2791:
2790:
2776:
2775:
2756:
2755:
2754:
2753:
2752:
2738:
2737:
2718:
2677:
2675:
2674:
2669:
2667:
2666:
2650:
2648:
2647:
2642:
2640:
2639:
2613:
2611:
2610:
2605:
2603:
2600:
2596:
2595:
2594:
2569:
2567:
2566:
2561:
2559:
2556:
2552:
2551:
2550:
2519:
2517:
2516:
2511:
2509:
2507:
2506:
2493:
2488:
2479:
2474:
2466:
2458:
2457:
2439:
2431:
2430:
2418:
2409:
2407:
2406:
2401:
2396:
2395:
2370:
2368:
2367:
2362:
2360:
2358:
2357:
2342:
2337:
2326:
2315:
2303:
2301:
2300:
2295:
2287:
2269:
2267:
2266:
2261:
2259:
2258:
2242:
2240:
2239:
2234:
2232:
2231:
2211:
2209:
2208:
2203:
2188:
2186:
2185:
2180:
2178:
2177:
2168:
2163:
2155:
2150:
2138:
2136:
2135:
2130:
2113:
2112:
2087:
2085:
2083:
2082:
2077:
2075:
2064:
2053:
2040:
2038:
2037:
2032:
2017:
2015:
2014:
2009:
2007:
2004:
2000:
1999:
1998:
1975:
1973:
1972:
1967:
1965:
1964:
1931:
1929:
1928:
1923:
1921:
1918:
1914:
1913:
1912:
1889:
1887:
1886:
1881:
1879:
1878:
1848:
1846:
1845:
1840:
1828:
1826:
1825:
1820:
1812:
1811:
1799:
1798:
1782:
1780:
1779:
1774:
1769:
1751:
1749:
1748:
1743:
1741:
1740:
1724:
1722:
1721:
1716:
1714:
1713:
1694:
1692:
1691:
1686:
1681:
1679:
1678:
1677:
1676:
1675:
1661:
1660:
1641:
1640:
1639:
1638:
1637:
1623:
1622:
1603:
1570:
1568:
1567:
1562:
1548:
1547:
1532:
1529:
1508:
1506:
1505:
1500:
1492:
1491:
1467:
1466:
1392:
1390:
1389:
1384:
1382:
1381:
1380:
1379:
1365:
1364:
1295:
1293:
1292:
1287:
1279:
1255:
1253:
1252:
1247:
1245:
1234:
1223:
1211:
1209:
1208:
1203:
1184:
1175:
1173:
1172:
1167:
1156:
1125:
1114:
1105:
1103:
1102:
1097:
1095:
1084:
1073:
1065:
1064:
1047:
1040:
1038:
1037:
1032:
1020:
1018:
1017:
1012:
1010:
1009:
990:
988:
987:
982:
953:
952:
918:
916:
915:
910:
898:
896:
895:
890:
872:
870:
869:
864:
816:
814:
813:
808:
796:
794:
793:
788:
752:
750:
749:
744:
732:
730:
729:
724:
712:
710:
709:
704:
702:
698:
647:
645:
644:
639:
637:
629:
621:
620:
601:
599:
598:
593:
591:
583:
567:
565:
564:
559:
543:
541:
540:
535:
523:
521:
520:
515:
503:
501:
500:
495:
493:
467:
444:
442:
441:
436:
424:
422:
421:
416:
379:frequency domain
350:
345:
344:
341:
340:
337:
334:
331:
328:
319:
317:[ˈboːdə]
310:
305:
304:
301:
300:
297:
294:
291:
288:
198:
193:
192:
189:
188:
185:
182:
179:
176:
138:High-pass filter
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
6255:
6254:
6250:
6249:
6248:
6246:
6245:
6244:
6195:
6194:
6180:
6175:
6174:
6167:
6151:
6144:
6137:
6120:
6116:
6109:
6093:
6089:
6082:
6066:
6062:
6055:
6041:
6037:
6020:
6016:
6007:
6005:
6000:
5999:
5995:
5985:
5981:
5974:
5952:
5948:
5943:
5938:
5937:
5932:
5925:
5918:
5914:
5897:
5893:
5883:
5879:
5874:
5842:
5835:
5828:
5819:
5812:
5776:
5768:Main articles:
5766:
5724:
5717:
5715:
5708:
5700:
5691:
5689:
5682:
5674:
5665:
5663:
5656:
5645:
5638:
5630:
5621:
5619:
5608:
5601:
5594:
5587:
5579:
5550:
5543:
5533:
5526:
5516:
5509:
5502:
5495:
5485:
5478:
5468:
5461:
5454:
5447:
5437:
5430:
5423:
5415:
5405:
5398:
5391:
5384:
5377:
5370:
5360:
5353:
5346:
5339:
5332:
5325:
5318:
5311:
5304:
5294:
5287:
5269:
5262:
5250:
5231:
5224:
5202:
5195:
5184:
5177:
5170:
5163:
5156:
5122:
5118:
5109:
5105:
5094:
5088:
5084:
5079:
5070:
5066:
5054:
5049:
5048:
5042:
5038:
5030:
5021:
5017:
5012:
5009:
5008:
5002:
4995:
4991:
4984:
4980:
4973:
4969:
4962:
4955:
4948:
4914:
4905:
4901:
4892:
4888:
4880:
4878:
4875:
4874:
4869:
4837:
4832:
4831:
4825:
4821:
4813:
4802:
4793:
4789:
4780:
4776:
4768:
4753:
4749:
4740:
4736:
4731:
4728:
4727:
4722:
4715:
4708:
4701:
4687:
4680:
4673:
4662:
4654:feedback factor
4643:
4612:
4608:
4598:
4592:
4588:
4586:
4577:
4573:
4571:
4568:
4567:
4553:
4547:
4540:
4537:
4528:
4525:
4516:
4513:
4504:
4501:
4478:
4456:
4452:
4450:
4447:
4446:
4429:
4425:
4420:
4415:
4412:
4411:
4408:
4406:Normalized plot
4386:
4382:
4374:
4371:
4370:
4353:
4349:
4344:
4339:
4336:
4335:
4318:
4314:
4312:
4309:
4308:
4292:
4289:
4288:
4272:
4269:
4268:
4246:
4242:
4237:
4225:
4221:
4204:
4195:
4191:
4183:
4180:
4179:
4173:
4147:
4143:
4141:
4138:
4137:
4110:
4106:
4101:
4081:
4078:
4077:
4060:
4056:
4051:
4046:
4043:
4042:
4025:
4021:
4019:
4016:
4015:
3995:
3991:
3986:
3981:
3978:
3977:
3960:
3956:
3954:
3951:
3950:
3926:
3923:
3922:
3905:
3904:
3888:
3883:
3873:
3869:
3867:
3860:
3856:
3835:
3834:
3821:
3817:
3812:
3807:
3800:
3796:
3775:
3774:
3759:
3755:
3750:
3745:
3738:
3734:
3729:
3711:
3710:
3705:
3694:
3685:
3681:
3676:
3660:
3654:
3650:
3646:
3644:
3641:
3640:
3620:
3616:
3614:
3611:
3610:
3603:
3570:
3566:
3561:
3556:
3549:
3544:
3530:
3521:
3517:
3515:
3512:
3511:
3505:
3479:
3475:
3470:
3467:
3466:
3449:
3445:
3440:
3437:
3436:
3399:
3396:
3395:
3373:
3368:
3362:
3358:
3353:
3351:
3348:
3347:
3331:
3325:
3321:
3316:
3308:
3305:
3304:
3287:
3283:
3278:
3275:
3274:
3233:
3230:
3229:
3213:
3207:
3203:
3198:
3190:
3187:
3186:
3164:
3159:
3153:
3149:
3144:
3142:
3139:
3138:
3122:
3116:
3112:
3107:
3099:
3096:
3095:
3078:
3074:
3069:
3066:
3065:
3024:
3021:
3020:
3004:
2998:
2994:
2989:
2981:
2978:
2977:
2955:
2952:
2951:
2932:
2929:
2928:
2923:pole and zero:
2875:
2849:
2847:
2821:
2818:
2817:
2786:
2782:
2781:
2777:
2771:
2767:
2757:
2748:
2744:
2743:
2739:
2733:
2729:
2719:
2717:
2694:
2691:
2690:
2684:
2662:
2658:
2656:
2653:
2652:
2635:
2631:
2629:
2626:
2625:
2599:
2590:
2586:
2581:
2578:
2577:
2555:
2546:
2542:
2537:
2534:
2533:
2526:
2502:
2498:
2489:
2484:
2478:
2462:
2453:
2449:
2435:
2426:
2422:
2417:
2415:
2412:
2411:
2391:
2387:
2376:
2373:
2372:
2353:
2349:
2341:
2333:
2322:
2311:
2309:
2306:
2305:
2283:
2275:
2272:
2271:
2254:
2250:
2248:
2245:
2244:
2227:
2223:
2221:
2218:
2217:
2197:
2194:
2193:
2173:
2169:
2162:
2149:
2144:
2141:
2140:
2108:
2104:
2099:
2096:
2095:
2071:
2060:
2049:
2047:
2044:
2043:
2042:
2026:
2023:
2022:
2003:
1994:
1990:
1985:
1982:
1981:
1960:
1956:
1948:
1945:
1944:
1917:
1908:
1904:
1899:
1896:
1895:
1874:
1870:
1862:
1859:
1858:
1834:
1831:
1830:
1807:
1803:
1794:
1790:
1788:
1785:
1784:
1765:
1757:
1754:
1753:
1752:are constants,
1736:
1732:
1730:
1727:
1726:
1709:
1705:
1703:
1700:
1699:
1671:
1667:
1666:
1662:
1656:
1652:
1642:
1633:
1629:
1628:
1624:
1618:
1614:
1604:
1602:
1579:
1576:
1575:
1543:
1539:
1528:
1526:
1523:
1522:
1519:
1487:
1483:
1462:
1458:
1408:
1405:
1404:
1398:zeros and poles
1375:
1371:
1370:
1366:
1360:
1356:
1324:
1321:
1320:
1302:
1275:
1261:
1258:
1257:
1241:
1230:
1219:
1217:
1214:
1213:
1197:
1194:
1193:
1152:
1132:
1129:
1128:
1091:
1080:
1069:
1060:
1056:
1054:
1051:
1050:
1026:
1023:
1022:
1005:
1001:
999:
996:
995:
948:
944:
927:
924:
923:
904:
901:
900:
881:
878:
877:
825:
822:
821:
802:
799:
798:
773:
770:
769:
759:
738:
735:
734:
718:
715:
714:
676:
672:
664:
661:
660:
653:Bode phase plot
633:
625:
616:
612:
607:
604:
603:
587:
579:
577:
574:
573:
553:
550:
549:
529:
526:
525:
509:
506:
505:
489:
463:
461:
458:
457:
430:
427:
426:
401:
398:
397:
387:
375:servomechanisms
348:
325:
321:
308:
285:
281:
278:Bode phase plot
250:
220:Bode phase plot
196:
173:
169:
150:Low-pass filter
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
6253:
6243:
6242:
6237:
6232:
6227:
6222:
6217:
6212:
6207:
6193:
6192:
6186:
6179:
6178:External links
6176:
6173:
6172:
6165:
6142:
6135:
6114:
6107:
6087:
6080:
6060:
6053:
6035:
6014:
5993:
5979:
5972:
5945:
5944:
5942:
5939:
5936:
5935:
5930:
5923:
5912:
5904:maximally flat
5891:
5876:
5875:
5873:
5870:
5869:
5868:
5863:
5858:
5853:
5848:
5841:
5838:
5837:
5836:
5829:
5822:
5820:
5813:
5806:
5765:
5762:
5723:
5720:
5719:
5718:
5713:
5706:
5701:
5694:
5692:
5687:
5680:
5675:
5668:
5666:
5661:
5657:where | β
5654:
5643:
5636:
5631:
5624:
5622:
5617:
5606:
5599:
5592:
5585:
5580:
5573:
5571:
5548:
5541:
5531:
5524:
5514:
5507:
5500:
5493:
5483:
5476:
5466:
5459:
5452:
5445:
5435:
5428:
5421:
5413:
5403:
5396:
5389:
5382:
5375:
5368:
5358:
5351:
5344:
5337:
5330:
5323:
5316:
5309:
5302:
5292:
5285:
5267:
5260:
5249:
5246:
5229:
5222:
5200:
5193:
5182:
5175:
5168:
5161:
5154:
5128:
5125:
5121:
5117:
5112:
5108:
5104:
5101:
5097:
5087:
5082:
5078:
5073:
5069:
5065:
5062:
5057:
5052:
5041:
5037:
5033:
5029:
5024:
5020:
5016:
5000:
4993:
4989:
4982:
4978:
4971:
4967:
4960:
4953:
4946:
4936:
4935:
4924:
4921:
4917:
4913:
4904:
4900:
4891:
4887:
4883:
4867:
4857:
4856:
4845:
4840:
4835:
4824:
4820:
4816:
4812:
4809:
4805:
4801:
4796:
4792:
4788:
4779:
4775:
4771:
4767:
4764:
4761:
4756:
4752:
4748:
4739:
4735:
4720:
4713:
4706:
4699:
4685:
4678:
4671:
4665:open-loop gain
4660:
4641:
4635:
4634:
4623:
4611:
4607:
4604:
4601:
4591:
4585:
4576:
4546:
4543:
4542:
4541:
4538:
4531:
4529:
4526:
4519:
4517:
4514:
4507:
4505:
4502:
4495:
4493:
4477:
4474:
4455:
4428:
4423:
4419:
4407:
4404:
4385:
4381:
4378:
4352:
4347:
4343:
4317:
4296:
4276:
4265:
4264:
4245:
4241:
4236:
4231:
4228:
4224:
4220:
4217:
4214:
4211:
4207:
4203:
4194:
4190:
4187:
4172:
4169:
4146:
4131:
4130:
4118:
4109:
4104:
4100:
4097:
4094:
4091:
4088:
4085:
4059:
4054:
4050:
4024:
4012:
3994:
3989:
3985:
3959:
3930:
3919:
3918:
3903:
3899:
3891:
3882:
3876:
3872:
3866:
3863:
3859:
3855:
3852:
3849:
3846:
3843:
3840:
3838:
3836:
3832:
3820:
3816:
3810:
3806:
3803:
3799:
3795:
3792:
3789:
3786:
3783:
3780:
3778:
3776:
3770:
3758:
3754:
3748:
3744:
3741:
3737:
3733:
3728:
3725:
3722:
3719:
3716:
3714:
3712:
3708:
3704:
3701:
3697:
3693:
3684:
3679:
3675:
3672:
3669:
3666:
3663:
3661:
3653:
3649:
3648:
3619:
3602:
3601:Magnitude plot
3599:
3595:
3594:
3583:
3569:
3565:
3559:
3555:
3552:
3548:
3543:
3540:
3537:
3533:
3529:
3520:
3504:
3501:
3500:
3499:
3496:
3482:
3478:
3474:
3452:
3448:
3444:
3433:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3392:
3380:
3376:
3371:
3365:
3361:
3356:
3334:
3328:
3324:
3319:
3315:
3312:
3290:
3286:
3282:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3216:
3210:
3206:
3201:
3197:
3194:
3183:
3171:
3167:
3162:
3156:
3152:
3147:
3125:
3119:
3115:
3110:
3106:
3103:
3081:
3077:
3073:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3007:
3001:
2997:
2992:
2988:
2985:
2974:
2971:
2959:
2948:
2936:
2917:
2916:
2905:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2811:
2810:
2799:
2789:
2785:
2780:
2774:
2770:
2766:
2763:
2760:
2751:
2747:
2742:
2736:
2732:
2728:
2725:
2722:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2683:
2680:
2665:
2661:
2638:
2634:
2622:
2621:
2618:
2593:
2589:
2585:
2574:
2549:
2545:
2541:
2525:
2522:
2505:
2501:
2497:
2492:
2487:
2483:
2477:
2472:
2469:
2465:
2461:
2456:
2452:
2448:
2445:
2442:
2438:
2434:
2429:
2425:
2421:
2399:
2394:
2390:
2386:
2383:
2380:
2356:
2352:
2348:
2345:
2340:
2336:
2332:
2329:
2325:
2321:
2318:
2314:
2293:
2290:
2286:
2282:
2279:
2257:
2253:
2230:
2226:
2201:
2176:
2172:
2166:
2161:
2158:
2153:
2148:
2128:
2125:
2122:
2119:
2116:
2111:
2107:
2103:
2092:
2091:
2088:
2074:
2070:
2067:
2063:
2059:
2056:
2052:
2030:
2019:
1997:
1993:
1989:
1963:
1959:
1955:
1952:
1937:
1911:
1907:
1903:
1877:
1873:
1869:
1866:
1838:
1818:
1815:
1810:
1806:
1802:
1797:
1793:
1772:
1768:
1764:
1761:
1739:
1735:
1712:
1708:
1696:
1695:
1684:
1674:
1670:
1665:
1659:
1655:
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675:
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668:
636:
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624:
619:
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611:
590:
586:
582:
557:
546:imaginary unit
533:
513:
492:
488:
485:
482:
479:
476:
473:
470:
466:
447:Laplace domain
434:
414:
411:
408:
405:
386:
383:
274:Bode gain plot
258:control theory
254:circuit theory
249:
246:
162:control theory
128:
127:
42:
40:
33:
26:
9:
6:
4:
3:
2:
6252:
6241:
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6191:
6187:
6185:
6182:
6181:
6168:
6166:0-387-25746-2
6162:
6158:
6157:
6149:
6147:
6138:
6136:9783540105657
6132:
6128:
6124:
6118:
6110:
6108:0-8493-8570-9
6104:
6100:
6099:
6091:
6083:
6081:0-521-83539-9
6077:
6073:
6072:
6064:
6056:
6054:0-470-01167-X
6050:
6046:
6039:
6032:
6028:
6024:
6018:
6003:
5997:
5989:
5983:
5975:
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5867:
5864:
5862:
5859:
5857:
5854:
5852:
5849:
5847:
5844:
5843:
5833:
5830:Figure 12: A
5826:
5821:
5817:
5814:Figure 11: A
5810:
5805:
5804:
5803:
5801:
5797:
5793:
5789:
5785:
5781:
5775:
5771:
5764:Related plots
5761:
5758:
5756:
5751:
5749:
5745:
5741:
5733:
5728:
5712:
5705:
5698:
5693:
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5616:
5612:
5605:
5598:
5591:
5584:
5577:
5572:
5569:
5568:
5567:
5565:
5564:step response
5561:
5560:rule of thumb
5557:
5556:step response
5552:
5547:
5540:
5535:
5530:
5523:
5518:
5513:
5506:
5499:
5492:
5487:
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5329:
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5315:
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5284:
5279:
5277:
5273:
5266:
5259:
5255:
5245:
5243:
5239:
5235:
5234:minimum phase
5228:
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5217:
5213:
5208:
5206:
5199:
5192:
5188:
5181:
5174:
5167:
5160:
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4902:
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4609:
4605:
4602:
4599:
4589:
4583:
4574:
4566:
4565:
4564:
4562:
4561:phase margins
4558:
4552:
4535:
4530:
4523:
4518:
4511:
4506:
4499:
4494:
4491:
4490:
4489:
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4350:
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4294:
4274:
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3476:
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3404:
3401:
3393:
3378:
3374:
3363:
3359:
3326:
3322:
3313:
3310:
3288:
3284:
3280:
3273:the slope by
3272:
3256:
3253:
3247:
3241:
3238:
3235:
3208:
3204:
3195:
3192:
3184:
3169:
3165:
3154:
3150:
3117:
3113:
3104:
3101:
3079:
3075:
3071:
3064:the slope by
3063:
3047:
3044:
3038:
3032:
3029:
3026:
2999:
2995:
2986:
2983:
2975:
2972:
2957:
2949:
2934:
2926:
2925:
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2922:
2903:
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2816:
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2723:
2714:
2711:
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2702:
2696:
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2688:
2687:
2679:
2663:
2659:
2636:
2632:
2619:
2616:
2591:
2587:
2583:
2575:
2572:
2547:
2543:
2539:
2531:
2530:
2529:
2521:
2503:
2499:
2495:
2490:
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2481:
2475:
2467:
2459:
2454:
2450:
2440:
2432:
2427:
2423:
2392:
2388:
2384:
2381:
2354:
2350:
2346:
2343:
2338:
2327:
2316:
2288:
2277:
2255:
2251:
2228:
2224:
2215:
2199:
2190:
2174:
2164:
2159:
2156:
2151:
2126:
2123:
2120:
2117:
2114:
2109:
2105:
2101:
2089:
2065:
2054:
2028:
2020:
1995:
1991:
1987:
1979:
1961:
1957:
1953:
1950:
1942:
1938:
1935:
1909:
1905:
1901:
1893:
1875:
1871:
1867:
1864:
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1836:
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1804:
1800:
1795:
1791:
1770:
1762:
1759:
1737:
1733:
1710:
1706:
1682:
1672:
1668:
1657:
1653:
1649:
1646:
1634:
1630:
1619:
1615:
1611:
1608:
1599:
1596:
1593:
1587:
1581:
1574:
1573:
1572:
1555:
1549:
1544:
1540:
1536:
1533:
1514:
1496:
1488:
1484:
1480:
1477:
1471:
1468:
1463:
1459:
1455:
1452:
1446:
1440:
1437:
1434:
1425:
1419:
1413:
1410:
1403:
1402:
1401:
1399:
1376:
1372:
1361:
1357:
1353:
1350:
1344:
1341:
1338:
1332:
1326:
1319:
1318:
1317:
1314:
1311:
1307:
1297:
1280:
1269:
1266:
1263:
1235:
1224:
1199:
1186:
1179:
1177:
1163:
1157:
1146:
1143:
1140:
1137:
1134:
1127:
1126:
1123:
1116:
1109:
1107:
1085:
1074:
1066:
1061:
1057:
1049:
1048:
1045:
1042:
1028:
1006:
1002:
978:
972:
969:
966:
963:
957:
954:
949:
945:
941:
935:
929:
922:
921:
920:
906:
883:
860:
854:
851:
845:
842:
839:
833:
827:
820:
819:
818:
804:
781:
775:
767:
762:
754:
740:
720:
699:
692:
689:
686:
683:
677:
673:
669:
666:
658:
654:
649:
630:
622:
617:
613:
609:
584:
571:
555:
547:
531:
511:
504:of frequency
483:
480:
477:
474:
468:
455:
450:
448:
432:
409:
403:
396:
392:
382:
380:
376:
372:
368:
364:
360:
355:
353:
352:
343:
318:
313:
312:
303:
279:
275:
271:
267:
263:
259:
255:
245:
243:
239:
238:approximation
236:
232:
227:
225:
221:
217:
213:
209:
205:
201:
200:
191:
167:
163:
159:
151:
146:
139:
134:
124:
121:
113:
110:December 2011
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
6155:
6126:
6117:
6097:
6090:
6070:
6063:
6044:
6038:
6022:
6017:
6006:. Retrieved
5996:
5982:
5957:
5949:
5927:
5920:
5915:
5903:
5899:
5894:
5887:Nyquist plot
5880:
5851:Phase margin
5832:Nichols plot
5816:Nyquist plot
5790:. These are
5788:Nichols plot
5784:Nyquist plot
5777:
5774:Nichols plot
5770:Nyquist plot
5759:
5752:
5740:oscilloscope
5737:
5722:Bode plotter
5710:
5703:
5684:
5677:
5658:
5651:
5647:
5640:
5633:
5614:
5610:
5603:
5596:
5589:
5582:
5553:
5545:
5538:
5536:
5528:
5521:
5519:
5511:
5504:
5497:
5490:
5488:
5480:
5473:
5471:
5463:
5456:
5449:
5442:
5440:
5432:
5425:
5418:
5410:
5408:
5400:
5393:
5386:
5379:
5372:
5365:
5363:
5355:
5348:
5341:
5334:
5327:
5320:
5313:
5306:
5299:
5297:
5289:
5282:
5280:
5271:
5264:
5257:
5253:
5251:
5238:Nyquist plot
5226:
5219:
5215:
5211:
5210:If a simple
5209:
5205:phase margin
5204:
5197:
5190:
5186:
5179:
5172:
5165:
5158:
5151:
5147:phase margin
5145:
5143:
5004:
4997:
4986:
4975:
4964:
4957:
4950:
4943:
4939:
4937:
4864:
4858:
4717:
4710:
4703:
4696:
4694:
4682:
4675:
4668:
4667:). The gain
4664:
4657:
4653:
4649:
4645:
4638:
4636:
4554:
4551:Phase margin
4486:
4482:
4479:
4409:
4266:
4174:
4132:
3942:
3920:
3604:
3596:
3506:
3270:
3061:
2920:
2918:
2812:
2685:
2623:
2614:
2570:
2527:
2213:
2191:
2093:
1977:
1940:
1891:
1854:
1697:
1520:
1511:
1395:
1315:
1303:
1191:
1180:
1121:
1110:
1043:
993:
875:
763:
760:
652:
650:
453:
451:
393:system with
388:
371:phase margin
356:
277:
273:
251:
228:
219:
211:
165:
155:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
5908:Butterworth
5664:| = 1.
5496:occurs at
5274:gain). See
5272:closed-loop
5005:gain margin
4940:gain margin
4165:attenuation
2018:per decade.
764:Consider a
367:gain margin
260:, engineer
224:phase shift
148:Figure 1B:
136:Figure 1A:
69:"Bode plot"
6199:Categories
6031:0801880572
6008:2013-10-07
5941:References
5431:for small
4549:See also:
4171:Phase plot
2216:a certain
1976:(a pole),
1890:(a zero),
1306:asymptotes
899:to a time
544:being the
385:Definition
359:amplifiers
235:asymptotic
80:newspapers
18:Bode plots
5800:log scale
5744:frequency
5655:0 dB
5566:article.
5532:0 dB
5525:0 dB
5515:0 dB
5501:0 dB
5467:0 dB
5446:0 dB
5383:0 dB
5369:0 dB
5359:0 dB
5352:0 dB
5324:0 dB
5254:open-loop
5223:0 dB
5201:0 dB
5183:0 dB
5162:0 dB
5124:−
5120:β
5116:
5100:−
5077:
5036:β
5028:
4886:β
4868:0 dB
4819:β
4811:−
4774:β
4766:−
4734:β
4707:0 dB
4606:β
4454:ω
4427:ω
4418:ω
4384:ω
4377:ω
4351:ω
4342:ω
4316:ω
4295:ω
4275:ω
4244:ω
4240:ω
4235:
4227:−
4219:−
4210:ω
4189:
4145:ω
4108:ω
4099:ω
4093:
4084:−
4058:ω
4049:ω
4023:ω
3993:ω
3984:ω
3958:ω
3943:two lines
3929:ω
3881:ω
3871:ω
3854:
3845:−
3819:ω
3815:ω
3794:
3785:−
3757:ω
3753:ω
3727:
3700:ω
3674:
3568:ω
3564:ω
3536:ω
3405:
3311:ω
3242:
3236:−
3193:ω
3185:At every
3102:ω
3033:
3027:−
2984:ω
2976:At every
2880:
2854:
2845:
2839:−
2824:φ
2765:−
2727:−
2715:∏
2617:the line.
2573:the line.
2500:ω
2468:ω
2460:−
2441:ω
2355:∗
2347:⋅
2328:ω
2289:ω
2200:ω
2066:ω
2029:ω
1951:ω
1865:ω
1771:ω
1650:−
1612:−
1600:∏
1550:
1481:−
1472:
1456:∑
1441:
1414:
1354:−
1345:∏
1281:ω
1267:
1236:ω
1200:ω
1158:ω
1144:
1135:φ
1086:ω
1029:φ
973:φ
964:ω
958:
887:∞
884:−
852:ω
846:
805:ω
741:ω
721:ω
693:ω
670:
623:
556:ω
512:ω
484:ω
266:Bell Labs
166:Bode plot
5840:See also
5786:and the
5782:are the
5748:feedback
5570:Examples
5244:theory.
4287:, where
3607:decibels
3271:decrease
3062:increase
2214:equal to
1978:decrease
1892:increase
570:decibels
363:feedback
248:Overview
216:decibels
5910:design.
5558:. As a
4996:from |β
4963:. If |β
4652:is the
4267:versus
3503:Example
3346:(e.g.,
3137:(e.g.,
548:). The
206:of the
202:) is a
94:scholar
6163:
6133:
6105:
6078:
6051:
6029:
5970:
5256:gain)
4656:, and
4637:where
2842:arctan
2597:
2553:
2001:
1943:where
1934:decade
1915:
1857:where
1829:, and
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