Bode plot

log-frequency axis, to appearance the system's abundance response. It is usually a aggregate of a Bode consequence plot, cogent the consequence of the abundance acknowledgment gain, and a Bode appearance plot, cogent the abundance acknowledgment appearance shift.

Overview

Among his several important contributions to ambit approach and ascendancy theory, architect Hendrik Wade Bode (1905–1982), while alive at Bell Labs in the United States in the 1930s, devised a simple but authentic adjustment for graphing accretion and phase-shift plots. These buck his name, Bode accretion artifice and Bode appearance artifice (pronounced Boh-dee in English, Bow-duh in Dutch).1

The consequence arbor of the Bode artifice is usually bidding as decibels of power, that is by the 20 log rule: 20 times the accepted (base 10) logarithm of the amplitude gain. With the consequence accretion getting logarithmic, Bode plots accomplish multiplication of magnitudes a simple amount of abacus distances on the blueprint (in decibels), since

A Bode appearance artifice is a blueprint of appearance against frequency, aswell advised on a log-frequency axis, usually acclimated in affiliation with the consequence plot, to appraise how abundant a arresting will be phase-shifted. For archetype a arresting declared by: Asin(ωt) may be attenuated but aswell phase-shifted. If the arrangement attenuates it by a agency x and appearance accouterment it by −Φ the arresting out of the arrangement will be (A/x) sin(ωt − Φ). The appearance about-face Φ is about a action of frequency.

Phase can aswell be added anon from the graphical values, a actuality that is mathematically bright if appearance is apparent as the abstract allotment of the circuitous logarithm of a circuitous gain.

In Amount 1(a), the Bode plots are apparent for the one-pole highpass clarify function:

where f is the abundance in Hz, and f1 is the pole position in Hz, f1 = 100 Hz in the figure. Application the rules for circuitous numbers, the consequence of this action is

while the appearance is:

Care accept to be taken that the afflicted departure is set up to acknowledgment degrees, not radians. On the Bode consequence plot, decibels are used, and the advised consequence is:



In Amount 1(b), the Bode plots are apparent for the one-pole lowpass clarify function:

Also apparent in Amount 1(a) and 1(b) are the straight-line approximations to the Bode plots that are acclimated in duke analysis, and declared later.

The consequence and appearance Bode plots can hardly be afflicted apart of anniversary added — alteration the amplitude acknowledgment of the arrangement will a lot of acceptable change the appearance characteristics and carnality versa. For minimum-phase systems the appearance and amplitude characteristics can be acquired from anniversary added with the use of the Hilbert transform.

If the alteration action is a rational action with absolute poles and zeros, afresh the Bode artifice can be approximated with beeline lines. These asymptotic approximations are alleged beeline band Bode plots or uncorrected Bode plots and are advantageous because they can be fatigued by duke afterward a few simple rules. Simple plots can even be predicted after cartoon them.

The approximation can be taken added by acclimation the amount at anniversary blow frequency. The artifice is afresh alleged a adapted Bode plot.

editRules for hand-made Bode plot

The apriorism of a Bode artifice is that one can accede the log of a action in the form:

as a sum of the logs of its poles and zeros:

This abstraction is acclimated absolutely in the adjustment for cartoon appearance diagrams. The adjustment for cartoon amplitude plots around uses this idea, but back the log of the amplitude of anniversary pole or aught consistently starts at aught and alone has one asymptote change (the beeline lines), the adjustment can be simplified.

editStraight-line amplitude plot

Amplitude decibels is usually done application the version. Accustomed a alteration action in the form

where and are constants, , , and H is the alteration function:

at every amount of s area (a zero), access the abruptness of the band by per decade.

at every amount of s area (a pole), abatement the abruptness of the band by per decade.

The antecedent amount of the blueprint depends on the boundaries. The antecedent point is begin by putting the antecedent angular abundance ω into the action and award |H(jω)|.

The antecedent abruptness of the action at the antecedent amount depends on the amount and adjustment of zeros and poles that are at ethics beneath the antecedent value, and are begin application the aboriginal two rules.

To handle irreducible 2nd adjustment polynomials, can, in abounding cases, be approximated as .

Note that zeros and poles appear if ω is according to a assertive or . This is because the action in catechism is the consequence of H(jω), and back it is a circuitous function, . Thus at any abode area there is a aught or pole involving the appellation , the consequence of that appellation is .

editCorrected amplitude plot

To absolute a straight-line amplitude plot:

at every zero, put a point aloft the line,

at every pole, put a point beneath the line,

draw a bland ambit through those credibility application the beeline ambit as asymptotes (lines which the ambit approaches).

Note that this alteration adjustment does not absorb how to handle circuitous ethics of or . In the case of an irreducible polynomial, the best way to absolute the artifice is to in fact account the consequence of the alteration action at the pole or aught agnate to the irreducible polynomial, and put that dot over or beneath the band at that pole or zero.

editStraight-line appearance plot

Given a alteration action in the aforementioned anatomy as above:

the abstraction is to draw abstracted plots for anniversary pole and zero, afresh add them up. The absolute appearance ambit is accustomed by .

To draw the appearance plot, for anniversary pole and zero:

if A is positive, alpha band (with aught slope) at 0 degrees

if A is negative, alpha band (with aught slope) at 180 degrees

at every (for abiding zeros – ), access the abruptness by degrees per decade, alpha one decade afore (E.g.: )

at every (for abiding poles – ), abatement the abruptness by degrees per decade, alpha one decade afore (E.g.: )

"unstable" (right bisected plane) poles and zeros () accept adverse behavior

flatten the abruptness afresh if the appearance has afflicted by degrees (for a zero) or degrees (for a pole),

After acute one band for anniversary pole or zero, add the ambit calm to access the final appearance plot; that is, the final appearance artifice is the superposition of anniversary beforehand appearance plot.

Examples

A acquiescent (unity canyon bandage gain) lowpass RC filter, for instance has the afterward alteration action bidding in the abundance domain:

From the alteration action it can be bent that the blow abundance point fc (in hertz) is at the frequency

or (equivalently) at

area is the angular blow abundance in radians per second.

The alteration action in agreement of the angular frequencies becomes:

The aloft blueprint is the normalized anatomy of the alteration function. The Bode artifice is apparent in Figure 1(b) above, and architecture of the straight-line approximation is discussed next.

editMagnitude plot

The consequence (in decibels) of the alteration action above, (normalized and adapted to angular abundance form), accustomed by the decibel accretion announcement :

when advised against ascribe abundance on a logarithmic scale, can be approximated by two curve and it forms the asymptotic (approximate) consequence Bode artifice of the alteration function:

for angular frequencies beneath it is a accumbent bandage at 0 dB back at low frequencies the appellation is baby and can be neglected, authoritative the decibel accretion blueprint aloft according to zero,

for angular frequencies aloft it is a bandage with a abruptness of −20 dB per decade back at top frequencies the appellation dominates and the decibel accretion announcement aloft simplifies to which is a beeline bandage with a abruptness of −20 dB per decade.

These two curve accommodated at the bend frequency. From the plot, it can be apparent that for frequencies able-bodied beneath the bend frequency, the ambit has an abrasion of 0 dB, agnate to a accord canyon bandage gain, i.e. the amplitude of the clarify achievement equals the amplitude of the input. Frequencies aloft the bend abundance are attenuated – the college the frequency, the college the attenuation.

editPhase plot

The appearance Bode artifice is acquired by acute the appearance bend of the alteration action accustomed by

versus , area and are the ascribe and blow angular frequencies respectively. For ascribe frequencies abundant lower than corner, the arrangement is baby and accordingly the appearance bend is abutting to zero. As the arrangement increases the complete amount of the appearance increases and becomes –45 degrees if . As the arrangement increases for ascribe frequencies abundant greater than the bend frequency, the appearance bend asymptotically approaches −90 degrees. The abundance calibration for the appearance artifice is logarithmic.

editNormalized plot

The accumbent abundance axis, in both the consequence and appearance plots, can be replaced by the normalized (nondimensional) abundance arrangement . In such a case the artifice is said to be normalized and units of the frequencies are no best acclimated back all ascribe frequencies are now bidding as multiples of the blow abundance .

An example with pole and zero

Figures 2-5 added allegorize architecture of Bode plots. This archetype with both a pole and a aught shows how to use superposition. To begin, the apparatus are presented separately.

Figure 2 shows the Bode consequence artifice for a aught and a low-pass pole, and compares the two with the Bode beeline band plots. The straight-line plots are accumbent up to the pole (zero) area and afresh bead (rise) at 20 dB/decade. The additional Figure 3 does the aforementioned for the phase. The appearance plots are accumbent up to a abundance agency of ten beneath the pole (zero) area and afresh bead (rise) at 45°/decade until the abundance is ten times college than the pole (zero) location. The plots afresh are afresh accumbent at college frequencies at a final, absolute appearance change of 90°.

Figure 4 and Figure 5 appearance how superposition (simple addition) of a pole and aught artifice is done. The Bode beeline band plots afresh are compared with the exact plots. The aught has been confused to college abundance than the pole to accomplish a added absorbing example. Notice in Figure 4 that the 20 dB/decade bead of the pole is arrested by the 20 dB/decade acceleration of the aught consistent in a accumbent consequence artifice for frequencies aloft the aught location. Notice in Figure 5 in the appearance artifice that the straight-line approximation is appealing almost in the arena area both pole and aught affect the phase. Notice aswell in Figure 5 that the ambit of frequencies area the appearance changes in the beeline band artifice is bound to frequencies a agency of ten aloft and beneath the pole (zero) location. Area the appearance of the pole and the aught both are present, the straight-line appearance artifice is accumbent because the 45°/decade bead of the pole is arrested by the overlapping 45°/decade acceleration of the aught in the bound ambit of frequencies area both are alive contributors to the phase.

Example with pole and zero

Figure 2: Bode consequence artifice for aught and low-pass pole; curves labeled "Bode" are the straight-line Bode plots

Figure 3: Bode appearance artifice for aught and low-pass pole; curves labeled "Bode" are the straight-line Bode plots

Figure 4: Bode consequence artifice for pole-zero combination; the area of the aught is ten times college than in Figures 2&3; curves labeled "Bode" are the straight-line Bode plots

Figure 5: Bode appearance artifice for pole-zero combination; the area of the aught is ten times college than in Figures 2&3; curves labeled "Bode" are the straight-line Bode plots

Gain margin and phase margin

Bode plots are acclimated to appraise the adherence of abrogating acknowledgment amplifiers by award the accretion and appearance margins of an amplifier. The angle of accretion and appearance allowance is based aloft the accretion announcement for a abrogating acknowledgment amplifier accustomed by

where AFB is the accretion of the amplifier with acknowledgment (the closed-loop gain), β is the acknowledgment agency and AOL is the accretion after acknowledgment (the open-loop gain). The accretion AOL is a circuitous action of frequency, with both consequence and phase.note 1 Examination of this affiliation shows the achievability of absolute accretion (interpreted as instability) if the artefact βAOL = −1. (That is, the consequence of βAOL is accord and its appearance is −180°, the alleged Barkhausen adherence criterion). Bode plots are acclimated to actuate just how abutting an amplifier comes to acceptable this condition.

Key to this assurance are two frequencies. The first, labeled actuality as f180, is the abundance area the open-loop accretion flips sign. The second, labeled actuality f0dB, is the abundance area the consequence of the artefact | β AOL | = 1 (in dB, consequence 1 is 0 dB). That is, abundance f180 is bent by the condition:

where vertical confined denote the consequence of a circuitous bulk (for example, | a + j b | =  a2 + b21/2 ), and abundance f0dB is bent by the condition:

One admeasurement of adjacency to alternation is the accretion margin. The Bode appearance artifice locates the abundance area the appearance of βAOL alcove −180°, denoted actuality as abundance f180. Application this frequency, the Bode consequence artifice finds the consequence of βAOL. If |βAOL|180 = 1, the amplifier is unstable, as mentioned. If |βAOL|180 < 1, alternation does not occur, and the break in dB of the consequence of |βAOL|180 from |βAOL| = 1 is alleged the accretion margin. Because a consequence of one is 0 dB, the accretion allowance is artlessly one of the agnate forms: 20 log10( |βAOL|180) = 20 log10( |AOL|180) − 20 log10( 1 / β ).

Another agnate admeasurement of adjacency to alternation is the appearance margin. The Bode consequence artifice locates the abundance area the consequence of |βAOL| alcove unity, denoted actuality as abundance f0dB. Application this frequency, the Bode appearance artifice finds the appearance of βAOL. If the appearance of βAOL( f0dB) > −180°, the alternation action cannot be met at any abundance (because its consequence is traveling to be < 1 if f = f180), and the ambit of the appearance at f0dB in degrees aloft −180° is alleged the appearance margin.

If a simple yes or no on the adherence affair is all that is needed, the amplifier is abiding if f0dB < f180. This archetype is acceptable to adumbrate adherence alone for amplifiers acceptable some restrictions on their pole and aught positions (minimum appearance systems). Although these restrictions usually are met, if they are not addition adjustment have to be used, such as the Nyquist plot.23

editExamples application Bode plots

Figures 6 and 7 allegorize the accretion behavior and terminology. For a three-pole amplifier, Figure 6 compares the Bode artifice for the accretion after acknowledgment (the open-loop gain) AOL with the accretion with acknowledgment AFB (the closed-loop gain). See abrogating acknowledgment amplifier for added detail.

In this example, AOL = 100 dB at low frequencies, and 1 / β = 58 dB. At low frequencies, AFB ≈ 58 dB as well.

Because the open-loop accretion AOL is advised and not the artefact β AOL, the action AOL = 1 / β decides f0dB. The acknowledgment accretion at low frequencies and for ample AOL is AFB ≈ 1 / β (look at the blueprint for the acknowledgment accretion at the alpha of this area for the case of ample accretion AOL), so an agnate way to acquisition f0dB is to attending area the acknowledgment accretion intersects the open-loop gain. (Frequency f0dB is bare after to acquisition the appearance margin.)

Near this crossover of the two assets at f0dB, the Barkhausen belief are about annoyed in this example, and the acknowledgment amplifier exhibits a massive aiguille in accretion (it would be beyond if β AOL = −1). Beyond the accord accretion abundance f0dB, the open-loop accretion is abundantly baby that AFB ≈ AOL (examine the blueprint at the alpha of this area for the case of baby AOL).

Figure 7 shows the agnate appearance comparison: the appearance of the acknowledgment amplifier is about aught out to the abundance f180 area the open-loop accretion has a appearance of −180°. In this vicinity, the appearance of the acknowledgment amplifier plunges abruptly bottomward to become about the aforementioned as the appearance of the open-loop amplifier. (Recall, AFB ≈ AOL for baby AOL.)

Comparing the labeled credibility in Figure 6 and Figure 7, it is apparent that the accord accretion abundance f0dB and the phase-flip abundance f180 are actual about according in this amplifier, f180 ≈ f0dB ≈ 3.332 kHz, which agency the accretion allowance and appearance allowance are about zero. The amplifier is ambiguous stable.

Figures 8 and 9 allegorize the accretion allowance and appearance allowance for a altered bulk of acknowledgment β. The acknowledgment agency is called abate than in Figure 6 or 7, affective the action | β AOL | = 1 to lower frequency. In this example, 1 / β = 77 dB, and at low frequencies AFB ≈ 77 dB as well.

Figure 8 shows the accretion plot. From Figure 8, the circle of 1 / β and AOL occurs at f0dB = 1 kHz. Notice that the aiguille in the accretion AFB abreast f0dB is about gone.note 24

Figure 9 is the appearance plot. Application the amount of f0dB = 1 kHz begin aloft from the consequence artifice of Figure 8, the open-loop appearance at f0dB is −135°, which is a appearance allowance of 45° aloft −180°.

Using Figure 9, for a appearance of −180° the amount of f180 = 3.332 kHz (the aforementioned aftereffect as begin earlier, of coursenote 3). The open-loop accretion from Figure 8 at f180 is 58 dB, and 1 / β = 77 dB, so the accretion allowance is 19 dB.

Stability is not the sole archetype for amplifier response, and in abounding applications a added acrimonious appeal than adherence is acceptable footfall response. As a aphorism of thumb, acceptable footfall acknowledgment requires a appearance allowance of at atomic 45°, and generally a allowance of over 70° is advocated, decidedly area basic aberration due to accomplishment tolerances is an issue.5 See aswell the altercation of appearance allowance in the footfall acknowledgment article.

Examples

Figure 6: Accretion of acknowledgment amplifier AFB in dB and agnate open-loop amplifier AOL. Parameter 1/β = 58 dB, and at low frequencies AFB ≈ 58 dB as well. The accretion allowance in this amplifier is about aught because | βAOL| = 1 occurs at about f = f180°.

Figure 7: Appearance of acknowledgment amplifier °AFB in degrees and agnate open-loop amplifier °AOL. The appearance allowance in this amplifier is about aught because the phase-flip occurs at about the accord accretion abundance f = f0dB area | βAOL| = 1.

Figure 8: Accretion of acknowledgment amplifier AFB in dB and agnate open-loop amplifier AOL. In this example, 1 / β = 77 dB. The accretion allowance in this amplifier is 19 dB.

Figure 9: Appearance of acknowledgment amplifier AFB in degrees and agnate open-loop amplifier AOL. The appearance allowance in this amplifier is 45°.

Bode plotter

The Bode actor is an cyberbanking apparatus akin an oscilloscope, which produces a Bode diagram, or a graph, of a circuit's voltage accretion or appearance about-face advised adjoin abundance in a acknowledgment ascendancy arrangement or a filter. An archetype of this is apparent in Figure 10. It is acutely advantageous for allegory and testing filters and the adherence of acknowledgment ascendancy systems, through the altitude of bend (cutoff) frequencies and accretion and appearance margins.

This is identical to the action performed by a agent arrangement analyzer, but the arrangement analyzer is about acclimated at abundant college frequencies.

For education/research purposes, acute Bode diagrams for accustomed alteration functions facilitates bigger compassionate and accepting faster after-effects (see alien links).

Related plots

Two accompanying plots that affectation the aforementioned abstracts in altered alike systems are the Nyquist artifice and the Nichols plot. These are parametric plots, with abundance as the ascribe and consequence and appearance of the abundance acknowledgment as the output. The Nyquist artifice displays these in arctic coordinates, with consequence mapping to ambit and appearance to altercation (angle). The Nichols artifice displays these in ellipsoidal coordinates, on the log scale.