4477:
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.
5565:
5616:
4511:
4523:
4487:
4499:
5660:
122:
5686:
134:
3905:
5814:
5798:
25:
3631:
4152:. 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
3900:{\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}}}
5564:
5615:
5873:
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
1501:
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.
5979:
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
4472:
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
1301:
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
4476:
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
4473:
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°.
4510:
4522:
2667:. 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.
3581:
2507:
5128:
4843:
4251:
5659:
5551:, 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
2903:
4621:
2797:
1682:
5685:
4486:
2358:
5405:
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,
4498:
1496:
4922:
1163:
4116:
3636:
978:
4391:. 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.
700:
2176:
2079:
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.
1093:
1380:
3256:
3047:
3502:
3419:
1558:
1283:
4432:
4389:
4356:
4063:
3998:
5461:
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
4434:. 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
4469:
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.
2073:
1243:
5922:
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.
2402:
362:
required to maintain stability under variations in circuit characteristics caused during manufacture or during operation. The principles developed were applied to design problems of
4516:
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
2291:
4663:
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
3487:
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.
1770:
2005:
1919:
860:
2601:
2557:
4999:
4528:
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
4459:
4321:
4150:
4028:
3963:
4718:
1816:
5716:
4170:
3333:
3215:
3124:
3006:
635:
491:
3623:
1963:
1877:
5749:
For education and research purposes, plotting Bode diagrams for given transfer functions facilitates better understanding and getting faster results (see external links).
3496:
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:
2126:
3378:
3169:
2397:
5735:
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
1028:
886:
4294:
4274:
3928:
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2199:
2028:
1199:
804:
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720:
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511:
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1739:
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1008:
784:
412:
589:
2957:
2934:
1836:
906:
531:
432:
2808:
5241:
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
5360:, 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
4558:
4358:
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
2681:
1566:
141:(1st-order, one-pole) Bode magnitude plot (top) and Bode phase plot (bottom). The red data curve is approximated by the straight black line.
129:(1st-order, one-pole) Bode magnitude plot (top) and Bode phase plot (bottom). The red data curve is approximated by the straight black line.
5849:
4399:
The horizontal frequency axis, in both the magnitude and phase plots, can be replaced by the normalized (nondimensional) frequency ratio
5221:. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions (
6085:
5990:
5813:
5787:, with magnitude mapping to radius and phase to argument (angle). The Nichols plot displays these in rectangular coordinates, on the
2296:
5543:
Stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good
5783:, with frequency as the input and magnitude and phase of the frequency response as the output. The Nyquist plot displays these in
4552:
of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by
1395:
5854:
5174:) > −180°, the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when
4865:
242:
2010:
The initial value of the graph depends on the boundaries. The initial point is found by putting the initial angular frequency
5960:
3598:) of the transfer function above (normalized and converted to angular-frequency form), given by the decibel gain expression
5398:
Figure 7 shows the corresponding phase comparison: the phase of the feedback amplifier is nearly zero out to the frequency
1119:
4068:
914:
89:
6228:
5225:
systems). Although these restrictions usually are met, if they are not, then another method must be used, such as the
61:
6223:
6153:
6123:
6095:
6068:
6041:
5635:. The phase margin in this amplifier is nearly zero because the phase-flip occurs at almost the unity gain frequency
5230:
651:
335:
295:
183:
108:
5458:≈ 3.332 kHz, which means the gain margin and phase margin are nearly zero. The amplifier is borderline stable.
3586:
The Bode plot is shown in Figure 1(b) above, and construction of the straight-line approximation is discussed next.
3576:{\displaystyle H_{\text{lp}}(\mathrm {j} \omega )={\frac {1}{1+\mathrm {j} {\frac {\omega }{\omega _{\text{c}}}}}}.}
2131:
1041:
5915:
does not change with a change in feedback factor; it is a property of the open-loop gain. The value of the gain at
4323:
are the input and cutoff angular frequencies respectively. For input frequencies much lower than corner, the ratio
2609:
Draw a smooth curve through those points using the straight lines as asymptotes (lines which the curve approaches).
1311:
5544:
3220:
3011:
3386:
1513:
1248:
68:
4492:
Figure 2: Bode magnitude plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots
4402:
4361:
4326:
4033:
3968:
2802:
the idea is to draw separate plots for each pole and zero, then add them up. The actual phase curve is given by
2502:{\displaystyle {\sqrt {(x_{n}+\mathrm {j} \omega )(x_{n}-\mathrm {j} \omega )}}={\sqrt {x_{n}^{2}+\omega ^{2}}}}
6218:
6208:
6019:
5844:
5329:≈ 1 / β (look at the formula for the feedback gain at the beginning of this section for the case of large gain
4678:
46:
5975:
Van
Valkenburg, M. E. University of Illinois at Urbana-Champaign, "In memoriam: Hendrik W. Bode (1905-1982)",
2034:
1293:
For many practical problems, the detailed Bode plots can be approximated with straight-line segments that are
1204:
6213:
4504:
Figure 3: Bode phase plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots
2262:
6203:
5123:{\displaystyle 20\log _{10}|\beta A_{\text{OL}}|_{180}=20\log _{10}|A_{\text{OL}}|-20\log _{10}\beta ^{-1}}
1744:
75:
4838:{\displaystyle \beta A_{\text{OL}}(f_{180})=-|\beta A_{\text{OL}}(f_{180})|=-|\beta A_{\text{OL}}|_{180},}
1972:
1886:
812:
6178:
5264:
4545:
2568:
2524:
645:
354:
for use in telephone networks. He developed the graphical design technique of the Bode plots to show the
5896:
5731:, which produces a Bode diagram, or a graph, of a circuit's voltage gain or phase shift plotted against
4437:
4299:
4246:{\displaystyle \arg H_{\text{lp}}(\mathrm {j} \omega )=-\tan ^{-1}{\frac {\omega }{\omega _{\text{c}}}}}
4128:
4006:
3941:
6198:
42:
1775:
57:
6193:
3295:
3177:
3086:
2968:
594:
448:
230:
5739:
control systems, through the measurement of corner (cutoff) frequencies and gain and phase margins.
5533:= 3.332 kHz (the same result as found earlier, of course). The open-loop gain from Figure 8 at
4681:). Bode plots are used to determine just how close an amplifier comes to satisfying this condition.
3601:
750:
This section illustrates that a Bode plot is a visualization of the frequency response of a system.
5952:
5834:
1935:
1849:
5797:
121:
5743:
2086:
133:
35:
5430:
Comparing the labeled points in Figure 6 and Figure 7, it is seen that the unity gain frequency
4996:. Because a magnitude of 1 is 0 dB, the gain margin is simply one of the equivalent forms:
3338:
3129:
4849:
2363:
742:-axis as the magnitude plot, but the value for the phase is plotted on a linear vertical axis.
146:
4164:
The phase Bode plot is obtained by plotting the phase angle of the transfer function given by
1285:. These quantities, thus, characterize the frequency response and are shown in the Bode plot.
1013:
868:
4279:
4259:
3913:
3457:
3427:
3265:
3056:
2184:
2013:
1184:
789:
725:
705:
540:
496:
5944:
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2616:
2235:
2208:
1717:
1690:
986:
5552:
760:
388:
8:
5945:
4000:
term is small and can be neglected, making the decibel gain equation above equal to zero.
1201:, the system responds at the same frequency with an output that is amplified by a factor
564:
2898:{\displaystyle \varphi (s)=-\arctan {\frac {\operatorname {Im} }{\operatorname {Re} }}.}
5768:
5679:. In this example, 1 / β = 77 dB. The gain margin in this amplifier is 19 dB.
5591:≈ 58 dB as well. The gain margin in this amplifier is nearly zero because | β
2942:
2919:
1922:
1821:
891:
516:
417:
258:
196:
82:
5468:| = 1 to lower frequency. In this example, 1 / β = 77 dB, and at low frequencies
2613:
Note that this correction method does not incorporate how to handle complex values of
1305:
The premise of a Bode plot is that one can consider the log of a function in the form
6149:
6119:
6091:
6064:
6037:
6015:
5956:
5784:
5720:
5516:= 1 kHz found above from the magnitude plot of Figure 8, the open-loop phase at
3934:, forming the asymptotic (approximate) magnitude Bode plot of the transfer function:
1298:
754:
383:
379:
250:
219:
2962:
If the sum of the number of unstable zeros and poles is odd, add 180° to that basis.
6111:
5780:
4974:< 1, instability does not occur, and the separation in dB of the magnitude of |β
4123:
367:
366:
and other feedback control systems. The Bode plot is an example of analysis in the
312:
272:
160:
126:
4691:, is the frequency where the open-loop gain flips sign. The second, labeled here
1386:
192:
138:
4030:
is a line with a slope of −20 dB per decade, since at high frequencies the
6172:
1297:
of the precise response. The effect of each of the terms of a multiple element
534:
435:
363:
305:
246:
150:
6012:
Between Human and
Machine: Feedback, Control, and Computing Before Cybernetics
5277:= 100 dB at low frequencies, and 1 / β = 58 dB. At low frequencies,
6187:
6057:
Thomas H. Lee (2004). "§14.6. Gain and Phase
Margins as Stability Measures".
5548:
5343:
is to look where the feedback gain intersects the open-loop gain. (Frequency
5222:
4616:{\displaystyle A_{\text{FB}}={\frac {A_{\text{OL}}}{1+\beta A_{\text{OL}}}},}
226:
212:
6063:(2nd ed.). Cambridge UK: Cambridge University Press. pp. 451–453.
5388:(examine the formula at the beginning of this section for the case of small
5875:
5839:
5820:
5804:
5776:
5772:
5762:
5758:
5728:
5478:
Figure 8 shows the gain plot. From Figure 8, the intersection of 1 / β and
5226:
5135:
4549:
4539:
2792:{\displaystyle H(s)=A\prod {\frac {(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}},}
1677:{\displaystyle H(s)=A\prod {\frac {(s-x_{n})^{a_{n}}}{(s-y_{n})^{b_{n}}}},}
359:
4684:
Key to this determination are two frequencies. The first, labeled here as
1302:
could be used to identify feasible ranges of parameters for a new design.
6143:
6058:
5746:, but the network analyzer is typically used at much higher frequencies.
5540:
is 58 dB, and 1 / β = 77 dB, so the gain margin is 19 dB.
5207:
on the stability issue is all that is needed, the amplifier is stable if
5139:. The Bode magnitude plot locates the frequency where the magnitude of |β
4153:
786:. Assume that the system is subject to a sinusoidal input with frequency
557:-axis of the magnitude plot is logarithmic and the magnitude is given in
355:
4945:. Using this frequency, the Bode magnitude plot finds the magnitude of β
223:
6084:
William S. Levine (1996). "§10.1. Specifications of
Control System".
5788:
5732:
2360:. Thus at any place where there is a zero or pole involving the term
1294:
347:
254:
4065:
term dominates, and the decibel gain expression above simplifies to
24:
5736:
351:
6090:(2nd ed.). Boca Raton FL: CRC Press/IEEE Press. p. 163.
4118:, which is a straight line with a slope of −20 dB per decade.
3595:
558:
204:
6087:
The control handbook: the electrical engineering handbook series
6036:. Chichester, West Sussex, England: John Wiley & Sons, Ltd.
5153:. Using this frequency, the Bode phase plot finds the phase of β
4931:. The Bode phase plot locates the frequency where the phase of β
3965:
is a horizontal line at 0 dB, since at low frequencies the
257:
in the 1930s, devised a simple but accurate method for graphing
2353:{\displaystyle |H(\mathrm {j} \omega )|={\sqrt {H\cdot H^{*}}}}
2259:. This is because the function in question is the magnitude of
6116:
Invariance and
Systems Theory: Algebraic and Geometric Aspects
5133:
Another equivalent measure of proximity to instability is the
5976:
5887:
The critical amount of feedback where the peak in the gain
5727:
The Bode plotter is an electronic instrument resembling an
5715:
1491:{\displaystyle \log(f(x))=\log(A)+\sum a_{n}\log(x-c_{n}).}
327:
321:
287:
281:
175:
169:
6148:. Dordrecht, The Netherlands: Springer. pp. 157–163.
5767:
Two related plots that display the same data in different
1560:
to define decibels. Given a transfer function in the form
648:, commonly expressed in degrees, of the transfer function
4917:{\displaystyle |\beta A_{\text{OL}}(f_{\text{0 dB}})|=1.}
4698:, is the frequency where the magnitude of the product |β
5229:. Optimal gain and phase margins may be computed using
6060:
The design of CMOS radio-frequency integrated circuits
5523:
is −135°, which is a phase margin of 45° above −180°.
3424:
Flatten the slope again when the phase has changed by
1033:
It can be shown that the magnitude of the response is
5584:. Parameter 1/β = 58 dB, and at low frequencies
5315:. The feedback gain at low frequencies and for large
5002:
4868:
4721:
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4405:
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Given a transfer function in the same form as above,
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1207:
1187:
1158:{\displaystyle \varphi =\arg H(\mathrm {j} \omega ).}
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1016:
989:
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6031:
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346:
Bode was faced with the problem of designing stable
324:
318:
284:
278:
172:
166:
4963:≥ 1, the amplifier is unstable, as mentioned. If |β
4111:{\displaystyle -20\log(\omega /\omega _{\text{c}})}
2959:
is negative, start line (with zero slope) at −180°.
315:
275:
163:
49:. Unsourced material may be challenged and removed.
16:
Graph of the frequency response of a control system
6025:
5908:The frequency where the open-loop gain flips sign
5526:Using Figure 9, for a phase of −180° the value of
5122:
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4837:
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1002:
973:{\displaystyle y(t)=y_{0}\sin(\omega t+\varphi ),}
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900:
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714:
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629:
583:
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5951:. Springer Science & Business Media. p.
5742:This is identical to the function performed by a
5698:in degrees and corresponding open-loop amplifier
5628:in degrees and corresponding open-loop amplifier
1181:In summary, subjected to an input with frequency
438:) consists of a magnitude plot and a phase plot.
6185:
6141:
6110:
5374:, the open-loop gain is sufficiently small that
4633:is the gain of the amplifier with feedback (the
4464:
3292:degrees per decade, beginning one decade before
3083:degrees per decade, beginning one decade before
2936:is positive, start line (with zero slope) at 0°.
5509:Figure 9 is the phase plot. Using the value of
5492:= 1 kHz. Notice that the peak in the gain
4938:reaches −180°, denoted here as frequency
4927:One measure of proximity to instability is the
4677:is unity and its phase is −180°, the so-called
4544:Bode plots are used to assess the stability of
4533:
3930:on a logarithmic scale, can be approximated by
3383:"Unstable" (right half-plane) poles and zeros (
1505:
1288:
865:that is applied persistently, i.e. from a time
722:. The phase is plotted on the same logarithmic
695:{\displaystyle \arg \left(H(s=j\omega )\right)}
199:of a system. It is usually a combination of a
4003:The second line for angular frequencies above
2171:{\displaystyle ({\sqrt {a}}x+{\sqrt {c}})^{2}}
1088:{\displaystyle y_{0}=|H(\mathrm {j} \omega )|}
983:i.e., also a sinusoidal signal with amplitude
6056:
5969:
5936:
5719:Figure 10: Amplitude diagram of a 10th-order
3938:The first line for angular frequencies below
2083:To handle irreducible 2nd-order polynomials,
1375:{\displaystyle f(x)=A\prod (x-c_{n})^{a_{n}}}
241:Among his several important contributions to
6032:Skogestad, Sigurd; Postlewaite, Ian (2005).
5705:. The phase margin in this amplifier is 45°.
5672:in dB and corresponding open-loop amplifier
5577:in dB and corresponding open-loop amplifier
5236:
3251:{\displaystyle -\operatorname {Re} (p)<0}
3042:{\displaystyle -\operatorname {Re} (z)<0}
261:and phase-shift plots. These bear his name,
6173:How to draw piecewise asymptotic Bode plots
5350:is needed later to find the phase margin.)
5192:in degrees above −180° is called the
5146:| reaches unity, denoted here as frequency
3414:{\displaystyle \operatorname {Re} (s)>0}
2670:
2517:To correct a straight-line amplitude plot:
2512:
1553:{\displaystyle {\text{dB}}=20\log _{10}(X)}
1278:{\displaystyle \arg H(\mathrm {j} \omega )}
4427:{\displaystyle \omega /\omega _{\text{c}}}
4384:{\displaystyle \omega =\omega _{\text{c}}}
4351:{\displaystyle \omega /\omega _{\text{c}}}
4058:{\displaystyle \omega /\omega _{\text{c}}}
3993:{\displaystyle \omega /\omega _{\text{c}}}
6137:
6135:
6104:
5444:are very nearly equal in this amplifier,
1510:Amplitude decibels is usually done using
109:Learn how and when to remove this message
5714:
5353:Near this crossover of the two gains at
2068:{\displaystyle |H(\mathrm {j} \omega )|}
1238:{\displaystyle |H(\mathrm {j} \omega )|}
132:
120:
6177:Gnuplot code for generating Bode plot:
5367:= −1). Beyond the unity gain frequency
2128:can, in many cases, be approximated as
203:, expressing the magnitude (usually in
6186:
6132:
5947:Analog and Digital Signals and Systems
5855:Electrochemical impedance spectroscopy
5850:Bode's magnitude (gain)–phase relation
5691:Figure 9: Phase of feedback amplifier
5621:Figure 7: Phase of feedback amplifier
4705:| = 1 = 0 dB. That is, frequency
2293:, and since it is a complex function,
2286:{\displaystyle H(\mathrm {j} \omega )}
2181:Note that zeros and poles happen when
303:, although the Dutch pronunciation is
6077:
5665:Figure 8: Gain of feedback amplifier
5570:Figure 6: Gain of feedback amplifier
1765:{\displaystyle s=\mathrm {j} \omega }
745:
304:
5878:have to be used to assess stability.
5823:of the same response from Figure 11.
5185:), and the distance of the phase at
3910:Then plotted versus input frequency
2000:{\displaystyle 20b_{n}\ {\text{dB}}}
1914:{\displaystyle 20a_{n}\ {\text{dB}}}
1113:
1035:
855:{\displaystyle u(t)=\sin(\omega t),}
47:adding citations to reliable sources
18:
5980:original Dutch be used as boh-dah."
2596:{\displaystyle 3b_{n}\ {\text{dB}}}
2552:{\displaystyle 3a_{n}\ {\text{dB}}}
908:. The response will be of the form
434:being the complex frequency in the
207:) of the frequency response, and a
13:
6050:
5991:"Vertaling van postbode, NL>EN"
4652:is the gain without feedback (the
4454:{\displaystyle \omega _{\text{c}}}
4394:
4316:{\displaystyle \omega _{\text{c}}}
4194:
4145:{\displaystyle \omega _{\text{c}}}
4023:{\displaystyle \omega _{\text{c}}}
3958:{\displaystyle \omega _{\text{c}}}
3797:
3735:
3684:
3546:
3520:
2452:
2425:
2312:
2273:
2050:
1755:
1265:
1220:
1142:
1070:
875:
561:, i.e., a value for the magnitude
14:
6240:
6166:
6118:. New York, NY: Springer-Verlag.
5294:is plotted and not the product β
4670:= −1 (that is, the magnitude of β
3589:
5812:
5796:
5752:
5684:
5658:
5614:
5563:
5336:), so an equivalent way to find
4521:
4509:
4497:
4485:
2399:, the magnitude of that term is
1811:{\displaystyle a_{n},b_{n}>0}
311:
271:
159:
23:
5902:
5881:
5710:
4859:is determined by the condition
4848:where vertical bars denote the
4712:is determined by the condition
3328:{\displaystyle \omega =|y_{n}|}
3210:{\displaystyle \omega =|y_{n}|}
3119:{\displaystyle \omega =|x_{n}|}
3001:{\displaystyle \omega =|x_{n}|}
630:{\displaystyle 20\log _{10}|H|}
486:{\displaystyle |H(s=j\omega )|}
34:needs additional citations for
6179:DIN-A4 printing template (pdf)
6034:Multivariable Feedback Control
6004:
5983:
5867:
5084:
5069:
5039:
5020:
4904:
4900:
4887:
4870:
4822:
4803:
4792:
4788:
4775:
4758:
4748:
4735:
4679:Barkhausen stability criterion
4201:
4190:
4105:
4084:
3695:
3691:
3680:
3666:
3618:{\displaystyle A_{\text{vdB}}}
3527:
3516:
3402:
3396:
3358:
3343:
3321:
3306:
3239:
3233:
3203:
3188:
3149:
3134:
3112:
3097:
3030:
3024:
2994:
2979:
2886:
2883:
2877:
2871:
2860:
2857:
2851:
2845:
2821:
2815:
2767:
2747:
2729:
2709:
2694:
2688:
2459:
2435:
2432:
2408:
2386:
2367:
2323:
2319:
2308:
2301:
2280:
2269:
2159:
2135:
2061:
2057:
2046:
2039:
2030:into the function and finding
1652:
1632:
1614:
1594:
1579:
1573:
1547:
1541:
1482:
1463:
1438:
1432:
1420:
1417:
1411:
1405:
1356:
1336:
1324:
1318:
1272:
1261:
1231:
1227:
1216:
1209:
1149:
1138:
1081:
1077:
1066:
1059:
964:
949:
927:
921:
846:
837:
825:
819:
773:
767:
757:system with transfer function
684:
669:
623:
615:
577:
569:
479:
475:
460:
453:
401:
395:
1:
5943:R. K. Rao Yarlagadda (2010).
5929:
5891:disappears altogether is the
5437:and the phase-flip frequency
5231:Nevanlinna–Pick interpolation
4850:magnitude of a complex number
4465:An example with zero and pole
4159:
1958:{\displaystyle \omega =y_{n}}
1872:{\displaystyle \omega =x_{n}}
445:is the graph of the function
373:
269:. "Bode" is often pronounced
222:in the 1930s, the plot is an
5723:plotted using a Bode plotter
5598:| = 1 occurs at almost
5252:with the gain with feedback
4546:negative-feedback amplifiers
4534:Gain margin and phase margin
4122:These two lines meet at the
2908:To draw the phase plot, for
1506:Straight-line amplitude plot
1385:as a sum of the logs of its
1289:Rules for handmade Bode plot
1111:and that the phase shift is
231:using straight line segments
7:
5845:Bode's sensitivity integral
5828:
5287:Because the open-loop gain
5265:negative feedback amplifier
2565:At every pole, put a point
2521:At every zero, put a point
2121:{\displaystyle ax^{2}+bx+c}
1171:
1101:
1030:with respect to the input.
236:
229:of the frequency response,
218:As originally conceived by
10:
6245:
5756:
4537:
4481:Example with pole and zero
3491:
3373:{\displaystyle |y_{n}|/10}
3164:{\displaystyle |x_{n}|/10}
1838:is the transfer function:
591:is plotted on the axis at
6229:Filter frequency response
6142:Willy M C Sansen (2006).
5237:Examples using Bode plots
3421:) have opposite behavior.
2392:{\displaystyle (s+x_{n})}
1969:the slope of the line by
1883:the slope of the line by
6224:Classical control theory
6145:Analog design essentials
5860:
5835:Analog signal processing
4548:by finding the gain and
3454:degrees (for a zero) or
2671:Straight-line phase plot
2513:Corrected amplitude plot
1023:{\displaystyle \varphi }
881:{\displaystyle -\infty }
5744:vector network analyzer
4289:{\displaystyle \omega }
4269:{\displaystyle \omega }
3923:{\displaystyle \omega }
3477:{\displaystyle 90b_{n}}
3447:{\displaystyle 90a_{n}}
3285:{\displaystyle 45b_{n}}
3076:{\displaystyle 45a_{n}}
2194:{\displaystyle \omega }
2023:{\displaystyle \omega }
1194:{\displaystyle \omega }
799:{\displaystyle \omega }
735:{\displaystyle \omega }
715:{\displaystyle \omega }
550:{\displaystyle \omega }
506:{\displaystyle \omega }
5724:
5475:≈ 77 dB as well.
5284:≈ 58 dB as well.
5124:
4918:
4839:
4617:
4455:
4428:
4385:
4352:
4317:
4290:
4270:
4247:
4146:
4112:
4059:
4024:
3994:
3959:
3924:
3901:
3619:
3577:
3478:
3448:
3415:
3374:
3329:
3286:
3252:
3211:
3165:
3120:
3077:
3043:
3002:
2953:
2930:
2899:
2793:
2661:
2634:
2597:
2553:
2503:
2393:
2354:
2287:
2253:
2226:
2195:
2172:
2122:
2069:
2024:
2001:
1959:
1915:
1873:
1832:
1812:
1766:
1735:
1708:
1678:
1554:
1492:
1376:
1279:
1239:
1195:
1159:
1089:
1024:
1004:
974:
902:
882:
856:
800:
780:
755:linear, time-invariant
736:
716:
696:
631:
585:
551:
527:
507:
487:
428:
408:
380:linear, time-invariant
147:electrical engineering
142:
130:
6219:Electrical parameters
6209:Electronic amplifiers
5718:
5125:
4919:
4840:
4618:
4456:
4429:
4386:
4353:
4318:
4291:
4271:
4248:
4147:
4113:
4060:
4025:
3995:
3960:
3925:
3902:
3620:
3578:
3484:degrees (for a pole).
3479:
3449:
3416:
3375:
3330:
3287:
3253:
3212:
3166:
3121:
3078:
3044:
3003:
2954:
2931:
2900:
2794:
2662:
2660:{\displaystyle y_{n}}
2635:
2633:{\displaystyle x_{n}}
2598:
2554:
2504:
2394:
2355:
2288:
2254:
2252:{\displaystyle y_{n}}
2227:
2225:{\displaystyle x_{n}}
2196:
2173:
2123:
2070:
2025:
2002:
1960:
1916:
1874:
1833:
1813:
1767:
1736:
1734:{\displaystyle y_{n}}
1709:
1707:{\displaystyle x_{n}}
1679:
1555:
1493:
1377:
1280:
1245:and phase-shifted by
1240:
1196:
1160:
1090:
1025:
1005:
1003:{\displaystyle y_{0}}
975:
903:
883:
857:
801:
781:
737:
717:
697:
632:
586:
552:
528:
508:
488:
429:
409:
136:
124:
6214:Electronics concepts
6022:, pp. 127–131.
5993:. mijnwoordenboek.nl
5000:
4992:| = 1 is called the
4866:
4719:
4559:
4438:
4403:
4362:
4327:
4300:
4280:
4260:
4171:
4129:
4069:
4034:
4007:
3969:
3942:
3914:
3632:
3602:
3503:
3458:
3428:
3387:
3339:
3296:
3266:
3221:
3178:
3130:
3087:
3057:
3012:
2969:
2943:
2920:
2809:
2682:
2644:
2617:
2569:
2525:
2403:
2364:
2297:
2263:
2236:
2209:
2185:
2132:
2087:
2035:
2014:
1973:
1936:
1887:
1850:
1822:
1776:
1745:
1718:
1691:
1567:
1514:
1396:
1312:
1249:
1205:
1185:
1120:
1042:
1014:
987:
915:
892:
869:
813:
790:
779:{\displaystyle H(s)}
761:
726:
706:
652:
644:is the graph of the
595:
565:
541:
517:
497:
449:
418:
407:{\displaystyle H(s)}
389:
378:The Bode plot for a
309:, closer to English
43:improve this article
6204:Electronic feedback
5160:. If the phase of β
3882:
2483:
1010:shifted by a phase
584:{\displaystyle |H|}
443:Bode magnitude plot
253:, while working at
201:Bode magnitude plot
5769:coordinate systems
5725:
5120:
4914:
4835:
4613:
4451:
4424:
4381:
4348:
4313:
4286:
4266:
4243:
4142:
4108:
4055:
4020:
3990:
3955:
3920:
3897:
3895:
3868:
3615:
3594:The magnitude (in
3573:
3474:
3444:
3411:
3370:
3325:
3282:
3248:
3217:(for stable poles
3207:
3161:
3116:
3073:
3039:
3008:(for stable zeros
2998:
2949:
2926:
2895:
2789:
2657:
2630:
2593:
2549:
2499:
2469:
2389:
2350:
2283:
2249:
2222:
2191:
2168:
2118:
2065:
2020:
1997:
1955:
1928:At every value of
1911:
1869:
1842:At every value of
1828:
1808:
1762:
1731:
1704:
1674:
1550:
1488:
1372:
1275:
1235:
1191:
1155:
1085:
1020:
1000:
970:
898:
878:
852:
796:
776:
746:Frequency response
732:
712:
692:
627:
581:
547:
523:
503:
483:
424:
404:
197:frequency response
143:
131:
6199:Signal processing
6114:(February 1981).
6014:JHU Press, 2004,
6010:David A. Mindell
5962:978-1-4419-0034-0
5785:polar coordinates
5721:electronic filter
5270:In this example,
5267:for more detail.
5080:
5034:
4897:
4884:
4817:
4772:
4732:
4608:
4604:
4584:
4569:
4448:
4421:
4378:
4345:
4310:
4241:
4238:
4187:
4139:
4102:
4052:
4017:
3987:
3952:
3883:
3875:
3816:
3813:
3761:
3754:
3751:
3677:
3646:
3612:
3568:
3565:
3562:
3513:
2952:{\displaystyle A}
2929:{\displaystyle A}
2890:
2784:
2591:
2587:
2547:
2543:
2497:
2462:
2348:
2156:
2143:
1995:
1991:
1909:
1905:
1831:{\displaystyle H}
1669:
1520:
1299:transfer function
1179:
1178:
1109:
1108:
901:{\displaystyle t}
702:as a function of
526:{\displaystyle j}
427:{\displaystyle s}
384:transfer function
251:Hendrik Wade Bode
220:Hendrik Wade Bode
211:, expressing the
119:
118:
111:
93:
6236:
6194:Plots (graphics)
6160:
6159:
6139:
6130:
6129:
6112:Allen Tannenbaum
6108:
6102:
6101:
6081:
6075:
6074:
6054:
6048:
6047:
6029:
6023:
6008:
6002:
6001:
5999:
5998:
5987:
5981:
5973:
5967:
5966:
5950:
5940:
5923:
5906:
5900:
5885:
5879:
5871:
5816:
5800:
5781:parametric plots
5688:
5662:
5618:
5567:
5506:is almost gone.
5308:= 1 / β decides
5301:, the condition
5129:
5127:
5126:
5121:
5119:
5118:
5103:
5102:
5087:
5082:
5081:
5078:
5072:
5064:
5063:
5048:
5047:
5042:
5036:
5035:
5032:
5023:
5015:
5014:
4923:
4921:
4920:
4915:
4907:
4899:
4898:
4895:
4886:
4885:
4882:
4873:
4852:, and frequency
4844:
4842:
4841:
4836:
4831:
4830:
4825:
4819:
4818:
4815:
4806:
4795:
4787:
4786:
4774:
4773:
4770:
4761:
4747:
4746:
4734:
4733:
4730:
4635:closed-loop gain
4622:
4620:
4619:
4614:
4609:
4607:
4606:
4605:
4602:
4586:
4585:
4582:
4576:
4571:
4570:
4567:
4525:
4513:
4501:
4489:
4460:
4458:
4457:
4452:
4450:
4449:
4446:
4433:
4431:
4430:
4425:
4423:
4422:
4419:
4413:
4390:
4388:
4387:
4382:
4380:
4379:
4376:
4357:
4355:
4354:
4349:
4347:
4346:
4343:
4337:
4322:
4320:
4319:
4314:
4312:
4311:
4308:
4295:
4293:
4292:
4287:
4275:
4273:
4272:
4267:
4252:
4250:
4249:
4244:
4242:
4240:
4239:
4236:
4227:
4222:
4221:
4197:
4189:
4188:
4185:
4151:
4149:
4148:
4143:
4141:
4140:
4137:
4124:corner frequency
4117:
4115:
4114:
4109:
4104:
4103:
4100:
4094:
4064:
4062:
4061:
4056:
4054:
4053:
4050:
4044:
4029:
4027:
4026:
4021:
4019:
4018:
4015:
3999:
3997:
3996:
3991:
3989:
3988:
3985:
3979:
3964:
3962:
3961:
3956:
3954:
3953:
3950:
3929:
3927:
3926:
3921:
3906:
3904:
3903:
3898:
3896:
3889:
3885:
3884:
3881:
3876:
3873:
3867:
3866:
3857:
3826:
3822:
3818:
3817:
3815:
3814:
3811:
3802:
3800:
3766:
3762:
3760:
3756:
3755:
3753:
3752:
3749:
3740:
3738:
3719:
3702:
3698:
3687:
3679:
3678:
3675:
3669:
3648:
3647:
3644:
3624:
3622:
3621:
3616:
3614:
3613:
3610:
3582:
3580:
3579:
3574:
3569:
3567:
3566:
3564:
3563:
3560:
3551:
3549:
3534:
3523:
3515:
3514:
3511:
3483:
3481:
3480:
3475:
3473:
3472:
3453:
3451:
3450:
3445:
3443:
3442:
3420:
3418:
3417:
3412:
3379:
3377:
3376:
3371:
3366:
3361:
3356:
3355:
3346:
3334:
3332:
3331:
3326:
3324:
3319:
3318:
3309:
3291:
3289:
3288:
3283:
3281:
3280:
3257:
3255:
3254:
3249:
3216:
3214:
3213:
3208:
3206:
3201:
3200:
3191:
3170:
3168:
3167:
3162:
3157:
3152:
3147:
3146:
3137:
3125:
3123:
3122:
3117:
3115:
3110:
3109:
3100:
3082:
3080:
3079:
3074:
3072:
3071:
3048:
3046:
3045:
3040:
3007:
3005:
3004:
2999:
2997:
2992:
2991:
2982:
2958:
2956:
2955:
2950:
2935:
2933:
2932:
2927:
2904:
2902:
2901:
2896:
2891:
2889:
2863:
2837:
2798:
2796:
2795:
2790:
2785:
2783:
2782:
2781:
2780:
2779:
2765:
2764:
2745:
2744:
2743:
2742:
2741:
2727:
2726:
2707:
2666:
2664:
2663:
2658:
2656:
2655:
2639:
2637:
2636:
2631:
2629:
2628:
2602:
2600:
2599:
2594:
2592:
2589:
2585:
2584:
2583:
2558:
2556:
2555:
2550:
2548:
2545:
2541:
2540:
2539:
2508:
2506:
2505:
2500:
2498:
2496:
2495:
2482:
2477:
2468:
2463:
2455:
2447:
2446:
2428:
2420:
2419:
2407:
2398:
2396:
2395:
2390:
2385:
2384:
2359:
2357:
2356:
2351:
2349:
2347:
2346:
2331:
2326:
2315:
2304:
2292:
2290:
2289:
2284:
2276:
2258:
2256:
2255:
2250:
2248:
2247:
2231:
2229:
2228:
2223:
2221:
2220:
2200:
2198:
2197:
2192:
2177:
2175:
2174:
2169:
2167:
2166:
2157:
2152:
2144:
2139:
2127:
2125:
2124:
2119:
2102:
2101:
2076:
2074:
2072:
2071:
2066:
2064:
2053:
2042:
2029:
2027:
2026:
2021:
2006:
2004:
2003:
1998:
1996:
1993:
1989:
1988:
1987:
1964:
1962:
1961:
1956:
1954:
1953:
1920:
1918:
1917:
1912:
1910:
1907:
1903:
1902:
1901:
1878:
1876:
1875:
1870:
1868:
1867:
1837:
1835:
1834:
1829:
1817:
1815:
1814:
1809:
1801:
1800:
1788:
1787:
1771:
1769:
1768:
1763:
1758:
1740:
1738:
1737:
1732:
1730:
1729:
1713:
1711:
1710:
1705:
1703:
1702:
1683:
1681:
1680:
1675:
1670:
1668:
1667:
1666:
1665:
1664:
1650:
1649:
1630:
1629:
1628:
1627:
1626:
1612:
1611:
1592:
1559:
1557:
1556:
1551:
1537:
1536:
1521:
1518:
1497:
1495:
1494:
1489:
1481:
1480:
1456:
1455:
1381:
1379:
1378:
1373:
1371:
1370:
1369:
1368:
1354:
1353:
1284:
1282:
1281:
1276:
1268:
1244:
1242:
1241:
1236:
1234:
1223:
1212:
1200:
1198:
1197:
1192:
1173:
1164:
1162:
1161:
1156:
1145:
1114:
1103:
1094:
1092:
1091:
1086:
1084:
1073:
1062:
1054:
1053:
1036:
1029:
1027:
1026:
1021:
1009:
1007:
1006:
1001:
999:
998:
979:
977:
976:
971:
942:
941:
907:
905:
904:
899:
887:
885:
884:
879:
861:
859:
858:
853:
805:
803:
802:
797:
785:
783:
782:
777:
741:
739:
738:
733:
721:
719:
718:
713:
701:
699:
698:
693:
691:
687:
636:
634:
633:
628:
626:
618:
610:
609:
590:
588:
587:
582:
580:
572:
556:
554:
553:
548:
532:
530:
529:
524:
512:
510:
509:
504:
492:
490:
489:
484:
482:
456:
433:
431:
430:
425:
413:
411:
410:
405:
368:frequency domain
339:
334:
333:
330:
329:
326:
323:
320:
317:
308:
306:[ˈboːdə]
299:
294:
293:
290:
289:
286:
283:
280:
277:
187:
182:
181:
178:
177:
174:
171:
168:
165:
127:High-pass filter
114:
107:
103:
100:
94:
92:
51:
27:
19:
6244:
6243:
6239:
6238:
6237:
6235:
6234:
6233:
6184:
6183:
6169:
6164:
6163:
6156:
6140:
6133:
6126:
6109:
6105:
6098:
6082:
6078:
6071:
6055:
6051:
6044:
6030:
6026:
6009:
6005:
5996:
5994:
5989:
5988:
5984:
5974:
5970:
5963:
5941:
5937:
5932:
5927:
5926:
5921:
5914:
5907:
5903:
5886:
5882:
5872:
5868:
5863:
5831:
5824:
5817:
5808:
5801:
5765:
5757:Main articles:
5755:
5713:
5706:
5704:
5697:
5689:
5680:
5678:
5671:
5663:
5654:
5652:
5645:
5634:
5627:
5619:
5610:
5608:
5597:
5590:
5583:
5576:
5568:
5539:
5532:
5522:
5515:
5505:
5498:
5491:
5484:
5474:
5467:
5457:
5450:
5443:
5436:
5426:
5419:
5412:
5404:
5394:
5387:
5380:
5373:
5366:
5359:
5349:
5342:
5335:
5328:
5321:
5314:
5307:
5300:
5293:
5283:
5276:
5258:
5251:
5239:
5220:
5213:
5191:
5184:
5173:
5166:
5159:
5152:
5145:
5111:
5107:
5098:
5094:
5083:
5077:
5073:
5068:
5059:
5055:
5043:
5038:
5037:
5031:
5027:
5019:
5010:
5006:
5001:
4998:
4997:
4991:
4984:
4980:
4973:
4969:
4962:
4958:
4951:
4944:
4937:
4903:
4894:
4890:
4881:
4877:
4869:
4867:
4864:
4863:
4858:
4826:
4821:
4820:
4814:
4810:
4802:
4791:
4782:
4778:
4769:
4765:
4757:
4742:
4738:
4729:
4725:
4720:
4717:
4716:
4711:
4704:
4697:
4690:
4676:
4669:
4662:
4651:
4643:feedback factor
4632:
4601:
4597:
4587:
4581:
4577:
4575:
4566:
4562:
4560:
4557:
4556:
4542:
4536:
4529:
4526:
4517:
4514:
4505:
4502:
4493:
4490:
4467:
4445:
4441:
4439:
4436:
4435:
4418:
4414:
4409:
4404:
4401:
4400:
4397:
4395:Normalized plot
4375:
4371:
4363:
4360:
4359:
4342:
4338:
4333:
4328:
4325:
4324:
4307:
4303:
4301:
4298:
4297:
4281:
4278:
4277:
4261:
4258:
4257:
4235:
4231:
4226:
4214:
4210:
4193:
4184:
4180:
4172:
4169:
4168:
4162:
4136:
4132:
4130:
4127:
4126:
4099:
4095:
4090:
4070:
4067:
4066:
4049:
4045:
4040:
4035:
4032:
4031:
4014:
4010:
4008:
4005:
4004:
3984:
3980:
3975:
3970:
3967:
3966:
3949:
3945:
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3940:
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3915:
3912:
3911:
3894:
3893:
3877:
3872:
3862:
3858:
3856:
3849:
3845:
3824:
3823:
3810:
3806:
3801:
3796:
3789:
3785:
3764:
3763:
3748:
3744:
3739:
3734:
3727:
3723:
3718:
3700:
3699:
3694:
3683:
3674:
3670:
3665:
3649:
3643:
3639:
3635:
3633:
3630:
3629:
3609:
3605:
3603:
3600:
3599:
3592:
3559:
3555:
3550:
3545:
3538:
3533:
3519:
3510:
3506:
3504:
3501:
3500:
3494:
3468:
3464:
3459:
3456:
3455:
3438:
3434:
3429:
3426:
3425:
3388:
3385:
3384:
3362:
3357:
3351:
3347:
3342:
3340:
3337:
3336:
3320:
3314:
3310:
3305:
3297:
3294:
3293:
3276:
3272:
3267:
3264:
3263:
3222:
3219:
3218:
3202:
3196:
3192:
3187:
3179:
3176:
3175:
3153:
3148:
3142:
3138:
3133:
3131:
3128:
3127:
3111:
3105:
3101:
3096:
3088:
3085:
3084:
3067:
3063:
3058:
3055:
3054:
3013:
3010:
3009:
2993:
2987:
2983:
2978:
2970:
2967:
2966:
2944:
2941:
2940:
2921:
2918:
2917:
2912:pole and zero:
2864:
2838:
2836:
2810:
2807:
2806:
2775:
2771:
2770:
2766:
2760:
2756:
2746:
2737:
2733:
2732:
2728:
2722:
2718:
2708:
2706:
2683:
2680:
2679:
2673:
2651:
2647:
2645:
2642:
2641:
2624:
2620:
2618:
2615:
2614:
2588:
2579:
2575:
2570:
2567:
2566:
2544:
2535:
2531:
2526:
2523:
2522:
2515:
2491:
2487:
2478:
2473:
2467:
2451:
2442:
2438:
2424:
2415:
2411:
2406:
2404:
2401:
2400:
2380:
2376:
2365:
2362:
2361:
2342:
2338:
2330:
2322:
2311:
2300:
2298:
2295:
2294:
2272:
2264:
2261:
2260:
2243:
2239:
2237:
2234:
2233:
2216:
2212:
2210:
2207:
2206:
2186:
2183:
2182:
2162:
2158:
2151:
2138:
2133:
2130:
2129:
2097:
2093:
2088:
2085:
2084:
2060:
2049:
2038:
2036:
2033:
2032:
2031:
2015:
2012:
2011:
1992:
1983:
1979:
1974:
1971:
1970:
1949:
1945:
1937:
1934:
1933:
1906:
1897:
1893:
1888:
1885:
1884:
1863:
1859:
1851:
1848:
1847:
1823:
1820:
1819:
1796:
1792:
1783:
1779:
1777:
1774:
1773:
1754:
1746:
1743:
1742:
1741:are constants,
1725:
1721:
1719:
1716:
1715:
1698:
1694:
1692:
1689:
1688:
1660:
1656:
1655:
1651:
1645:
1641:
1631:
1622:
1618:
1617:
1613:
1607:
1603:
1593:
1591:
1568:
1565:
1564:
1532:
1528:
1517:
1515:
1512:
1511:
1508:
1476:
1472:
1451:
1447:
1397:
1394:
1393:
1387:zeros and poles
1364:
1360:
1359:
1355:
1349:
1345:
1313:
1310:
1309:
1291:
1264:
1250:
1247:
1246:
1230:
1219:
1208:
1206:
1203:
1202:
1186:
1183:
1182:
1141:
1121:
1118:
1117:
1080:
1069:
1058:
1049:
1045:
1043:
1040:
1039:
1015:
1012:
1011:
994:
990:
988:
985:
984:
937:
933:
916:
913:
912:
893:
890:
889:
870:
867:
866:
814:
811:
810:
791:
788:
787:
762:
759:
758:
748:
727:
724:
723:
707:
704:
703:
665:
661:
653:
650:
649:
642:Bode phase plot
622:
614:
605:
601:
596:
593:
592:
576:
568:
566:
563:
562:
542:
539:
538:
518:
515:
514:
498:
495:
494:
478:
452:
450:
447:
446:
419:
416:
415:
390:
387:
386:
376:
364:servomechanisms
337:
314:
310:
297:
274:
270:
267:Bode phase plot
239:
209:Bode phase plot
185:
162:
158:
139:Low-pass filter
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
6242:
6232:
6231:
6226:
6221:
6216:
6211:
6206:
6201:
6196:
6182:
6181:
6175:
6168:
6167:External links
6165:
6162:
6161:
6154:
6131:
6124:
6103:
6096:
6076:
6069:
6049:
6042:
6024:
6003:
5982:
5968:
5961:
5934:
5933:
5931:
5928:
5925:
5924:
5919:
5912:
5901:
5893:maximally flat
5880:
5865:
5864:
5862:
5859:
5858:
5857:
5852:
5847:
5842:
5837:
5830:
5827:
5826:
5825:
5818:
5811:
5809:
5802:
5795:
5754:
5751:
5712:
5709:
5708:
5707:
5702:
5695:
5690:
5683:
5681:
5676:
5669:
5664:
5657:
5655:
5650:
5646:where | β
5643:
5632:
5625:
5620:
5613:
5611:
5606:
5595:
5588:
5581:
5574:
5569:
5562:
5560:
5537:
5530:
5520:
5513:
5503:
5496:
5489:
5482:
5472:
5465:
5455:
5448:
5441:
5434:
5424:
5417:
5410:
5402:
5392:
5385:
5378:
5371:
5364:
5357:
5347:
5340:
5333:
5326:
5319:
5312:
5305:
5298:
5291:
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5274:
5256:
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5211:
5189:
5182:
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5150:
5143:
5117:
5114:
5110:
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5054:
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5018:
5013:
5009:
5005:
4989:
4982:
4978:
4971:
4967:
4960:
4956:
4949:
4942:
4935:
4925:
4924:
4913:
4910:
4906:
4902:
4893:
4889:
4880:
4876:
4872:
4856:
4846:
4845:
4834:
4829:
4824:
4813:
4809:
4805:
4801:
4798:
4794:
4790:
4785:
4781:
4777:
4768:
4764:
4760:
4756:
4753:
4750:
4745:
4741:
4737:
4728:
4724:
4709:
4702:
4695:
4688:
4674:
4667:
4660:
4654:open-loop gain
4649:
4630:
4624:
4623:
4612:
4600:
4596:
4593:
4590:
4580:
4574:
4565:
4535:
4532:
4531:
4530:
4527:
4520:
4518:
4515:
4508:
4506:
4503:
4496:
4494:
4491:
4484:
4482:
4466:
4463:
4444:
4417:
4412:
4408:
4396:
4393:
4374:
4370:
4367:
4341:
4336:
4332:
4306:
4285:
4265:
4254:
4253:
4234:
4230:
4225:
4220:
4217:
4213:
4209:
4206:
4203:
4200:
4196:
4192:
4183:
4179:
4176:
4161:
4158:
4135:
4120:
4119:
4107:
4098:
4093:
4089:
4086:
4083:
4080:
4077:
4074:
4048:
4043:
4039:
4013:
4001:
3983:
3978:
3974:
3948:
3919:
3908:
3907:
3892:
3888:
3880:
3871:
3865:
3861:
3855:
3852:
3848:
3844:
3841:
3838:
3835:
3832:
3829:
3827:
3825:
3821:
3809:
3805:
3799:
3795:
3792:
3788:
3784:
3781:
3778:
3775:
3772:
3769:
3767:
3765:
3759:
3747:
3743:
3737:
3733:
3730:
3726:
3722:
3717:
3714:
3711:
3708:
3705:
3703:
3701:
3697:
3693:
3690:
3686:
3682:
3673:
3668:
3664:
3661:
3658:
3655:
3652:
3650:
3642:
3638:
3637:
3608:
3591:
3590:Magnitude plot
3588:
3584:
3583:
3572:
3558:
3554:
3548:
3544:
3541:
3537:
3532:
3529:
3526:
3522:
3518:
3509:
3493:
3490:
3489:
3488:
3485:
3471:
3467:
3463:
3441:
3437:
3433:
3422:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3381:
3369:
3365:
3360:
3354:
3350:
3345:
3323:
3317:
3313:
3308:
3304:
3301:
3279:
3275:
3271:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3205:
3199:
3195:
3190:
3186:
3183:
3172:
3160:
3156:
3151:
3145:
3141:
3136:
3114:
3108:
3104:
3099:
3095:
3092:
3070:
3066:
3062:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
2996:
2990:
2986:
2981:
2977:
2974:
2963:
2960:
2948:
2937:
2925:
2906:
2905:
2894:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2800:
2799:
2788:
2778:
2774:
2769:
2763:
2759:
2755:
2752:
2749:
2740:
2736:
2731:
2725:
2721:
2717:
2714:
2711:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2672:
2669:
2654:
2650:
2627:
2623:
2611:
2610:
2607:
2582:
2578:
2574:
2563:
2538:
2534:
2530:
2514:
2511:
2494:
2490:
2486:
2481:
2476:
2472:
2466:
2461:
2458:
2454:
2450:
2445:
2441:
2437:
2434:
2431:
2427:
2423:
2418:
2414:
2410:
2388:
2383:
2379:
2375:
2372:
2369:
2345:
2341:
2337:
2334:
2329:
2325:
2321:
2318:
2314:
2310:
2307:
2303:
2282:
2279:
2275:
2271:
2268:
2246:
2242:
2219:
2215:
2190:
2165:
2161:
2155:
2150:
2147:
2142:
2137:
2117:
2114:
2111:
2108:
2105:
2100:
2096:
2092:
2081:
2080:
2077:
2063:
2059:
2056:
2052:
2048:
2045:
2041:
2019:
2008:
1986:
1982:
1978:
1952:
1948:
1944:
1941:
1926:
1900:
1896:
1892:
1866:
1862:
1858:
1855:
1827:
1807:
1804:
1799:
1795:
1791:
1786:
1782:
1761:
1757:
1753:
1750:
1728:
1724:
1701:
1697:
1685:
1684:
1673:
1663:
1659:
1654:
1648:
1644:
1640:
1637:
1634:
1625:
1621:
1616:
1610:
1606:
1602:
1599:
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613:
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579:
575:
571:
546:
535:imaginary unit
522:
502:
481:
477:
474:
471:
468:
465:
462:
459:
455:
436:Laplace domain
423:
403:
400:
397:
394:
375:
372:
263:Bode gain plot
247:control theory
243:circuit theory
238:
235:
151:control theory
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
6241:
6230:
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6171:
6170:
6157:
6155:0-387-25746-2
6151:
6147:
6146:
6138:
6136:
6127:
6125:9783540105657
6121:
6117:
6113:
6107:
6099:
6097:0-8493-8570-9
6093:
6089:
6088:
6080:
6072:
6070:0-521-83539-9
6066:
6062:
6061:
6053:
6045:
6043:0-470-01167-X
6039:
6035:
6028:
6021:
6017:
6013:
6007:
5992:
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5853:
5851:
5848:
5846:
5843:
5841:
5838:
5836:
5833:
5832:
5822:
5819:Figure 12: A
5815:
5810:
5806:
5803:Figure 11: A
5799:
5794:
5793:
5792:
5790:
5786:
5782:
5778:
5774:
5770:
5764:
5760:
5753:Related plots
5750:
5747:
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5694:
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5587:
5580:
5573:
5566:
5561:
5558:
5557:
5556:
5554:
5553:step response
5550:
5549:rule of thumb
5546:
5545:step response
5541:
5536:
5529:
5524:
5519:
5512:
5507:
5502:
5495:
5488:
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5273:
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5224:
5223:minimum phase
5217:
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4553:
4551:
4550:phase margins
4547:
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4507:
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3367:
3363:
3352:
3348:
3315:
3311:
3302:
3299:
3277:
3273:
3269:
3262:the slope by
3261:
3245:
3242:
3236:
3230:
3227:
3224:
3197:
3193:
3184:
3181:
3173:
3158:
3154:
3143:
3139:
3106:
3102:
3093:
3090:
3068:
3064:
3060:
3053:the slope by
3052:
3036:
3033:
3027:
3021:
3018:
3015:
2988:
2984:
2975:
2972:
2964:
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2240:
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2188:
2179:
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2140:
2115:
2112:
2109:
2106:
2103:
2098:
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2090:
2078:
2054:
2043:
2017:
2009:
1984:
1980:
1976:
1968:
1950:
1946:
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1503:
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1477:
1473:
1469:
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1435:
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1423:
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1408:
1402:
1399:
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1391:
1390:
1388:
1365:
1361:
1350:
1346:
1342:
1339:
1333:
1330:
1327:
1321:
1315:
1308:
1307:
1306:
1303:
1300:
1296:
1286:
1269:
1258:
1255:
1252:
1224:
1213:
1188:
1175:
1168:
1166:
1152:
1146:
1135:
1132:
1129:
1126:
1123:
1116:
1115:
1112:
1105:
1098:
1096:
1074:
1063:
1055:
1050:
1046:
1038:
1037:
1034:
1031:
1017:
995:
991:
967:
961:
958:
955:
952:
946:
943:
938:
934:
930:
924:
918:
911:
910:
909:
895:
872:
849:
843:
840:
834:
831:
828:
822:
816:
809:
808:
807:
793:
770:
764:
756:
751:
743:
729:
709:
688:
681:
678:
675:
672:
666:
662:
658:
655:
647:
643:
638:
619:
611:
606:
602:
598:
573:
560:
544:
536:
520:
500:
493:of frequency
472:
469:
466:
463:
457:
444:
439:
437:
421:
398:
392:
385:
381:
371:
369:
365:
361:
357:
353:
349:
344:
342:
341:
332:
307:
302:
301:
292:
268:
264:
260:
256:
252:
248:
244:
234:
232:
228:
227:approximation
225:
221:
216:
214:
210:
206:
202:
198:
194:
190:
189:
180:
156:
152:
148:
140:
135:
128:
123:
113:
110:
102:
99:December 2011
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
6144:
6115:
6106:
6086:
6079:
6059:
6052:
6033:
6027:
6011:
6006:
5995:. Retrieved
5985:
5971:
5946:
5938:
5916:
5909:
5904:
5892:
5888:
5883:
5876:Nyquist plot
5869:
5840:Phase margin
5821:Nichols plot
5805:Nyquist plot
5779:. These are
5777:Nichols plot
5773:Nyquist plot
5766:
5763:Nichols plot
5759:Nyquist plot
5748:
5741:
5729:oscilloscope
5726:
5711:Bode plotter
5699:
5692:
5673:
5666:
5647:
5640:
5636:
5629:
5622:
5603:
5599:
5592:
5585:
5578:
5571:
5542:
5534:
5527:
5525:
5517:
5510:
5508:
5500:
5493:
5486:
5479:
5477:
5469:
5462:
5460:
5452:
5445:
5438:
5431:
5429:
5421:
5414:
5407:
5399:
5397:
5389:
5382:
5375:
5368:
5361:
5354:
5352:
5344:
5337:
5330:
5323:
5316:
5309:
5302:
5295:
5288:
5286:
5278:
5271:
5269:
5260:
5253:
5246:
5242:
5240:
5227:Nyquist plot
5215:
5208:
5204:
5200:
5199:If a simple
5198:
5194:phase margin
5193:
5186:
5179:
5175:
5168:
5161:
5154:
5147:
5140:
5136:phase margin
5134:
5132:
4993:
4986:
4975:
4964:
4953:
4946:
4939:
4932:
4928:
4926:
4853:
4847:
4706:
4699:
4692:
4685:
4683:
4671:
4664:
4657:
4656:). The gain
4653:
4646:
4642:
4638:
4634:
4627:
4625:
4543:
4540:Phase margin
4475:
4471:
4468:
4398:
4255:
4163:
4121:
3931:
3909:
3593:
3585:
3495:
3259:
3050:
2909:
2907:
2801:
2674:
2612:
2603:
2559:
2516:
2202:
2180:
2082:
1966:
1929:
1880:
1843:
1686:
1509:
1500:
1384:
1304:
1292:
1180:
1169:
1110:
1099:
1032:
982:
864:
752:
749:
641:
639:
442:
440:
382:system with
377:
360:phase margin
345:
266:
262:
240:
217:
208:
200:
154:
144:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
5897:Butterworth
5653:| = 1.
5485:occurs at
5263:gain). See
5261:closed-loop
4994:gain margin
4929:gain margin
4154:attenuation
2007:per decade.
753:Consider a
356:gain margin
249:, engineer
213:phase shift
137:Figure 1B:
125:Figure 1A:
58:"Bode plot"
6188:Categories
6020:0801880572
5997:2013-10-07
5930:References
5420:for small
4538:See also:
4160:Phase plot
2205:a certain
1965:(a pole),
1879:(a zero),
1295:asymptotes
888:to a time
533:being the
374:Definition
348:amplifiers
224:asymptotic
69:newspapers
5789:log scale
5733:frequency
5644:0 dB
5555:article.
5521:0 dB
5514:0 dB
5504:0 dB
5490:0 dB
5456:0 dB
5435:0 dB
5372:0 dB
5358:0 dB
5348:0 dB
5341:0 dB
5313:0 dB
5243:open-loop
5212:0 dB
5190:0 dB
5172:0 dB
5151:0 dB
5113:−
5109:β
5105:
5089:−
5066:
5025:β
5017:
4875:β
4857:0 dB
4808:β
4800:−
4763:β
4755:−
4723:β
4696:0 dB
4595:β
4443:ω
4416:ω
4407:ω
4373:ω
4366:ω
4340:ω
4331:ω
4305:ω
4284:ω
4264:ω
4233:ω
4229:ω
4224:
4216:−
4208:−
4199:ω
4178:
4134:ω
4097:ω
4088:ω
4082:
4073:−
4047:ω
4038:ω
4012:ω
3982:ω
3973:ω
3947:ω
3932:two lines
3918:ω
3870:ω
3860:ω
3843:
3834:−
3808:ω
3804:ω
3783:
3774:−
3746:ω
3742:ω
3716:
3689:ω
3663:
3557:ω
3553:ω
3525:ω
3394:
3300:ω
3231:
3225:−
3182:ω
3174:At every
3091:ω
3022:
3016:−
2973:ω
2965:At every
2869:
2843:
2834:
2828:−
2813:φ
2754:−
2716:−
2704:∏
2606:the line.
2562:the line.
2489:ω
2457:ω
2449:−
2430:ω
2344:∗
2336:⋅
2317:ω
2278:ω
2189:ω
2055:ω
2018:ω
1940:ω
1854:ω
1760:ω
1639:−
1601:−
1589:∏
1539:
1470:−
1461:
1445:∑
1430:
1403:
1343:−
1334:∏
1270:ω
1256:
1225:ω
1189:ω
1147:ω
1133:
1124:φ
1075:ω
1018:φ
962:φ
953:ω
947:
876:∞
873:−
841:ω
835:
794:ω
730:ω
710:ω
682:ω
659:
612:
545:ω
501:ω
473:ω
255:Bell Labs
155:Bode plot
5829:See also
5775:and the
5771:are the
5737:feedback
5559:Examples
5233:theory.
4276:, where
3596:decibels
3260:decrease
3051:increase
2203:equal to
1967:decrease
1881:increase
559:decibels
352:feedback
237:Overview
205:decibels
5899:design.
5547:. As a
4985:from |β
4952:. If |β
4641:is the
4256:versus
3492:Example
3335:(e.g.,
3126:(e.g.,
537:). The
195:of the
191:) is a
83:scholar
6152:
6122:
6094:
6067:
6040:
6018:
5959:
5245:gain)
4645:, and
4626:where
2831:arctan
2586:
2542:
1990:
1932:where
1923:decade
1904:
1846:where
1818:, and
1687:where
513:(with
85:
78:
71:
64:
56:
5861:Notes
5499:near
5259:(the
5214:<
2604:below
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