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°.