Name: bode plot slope and phase changes
Uploaded: 2025-05-22T09:06:14.000Z
Duration: 1 min 23 s
Channel: JoVE

Considérons la fonction de transfert dans sa forme standard, avec des pôles et des zéros
Pour une fonction de transfert avec un zéro simple, le gain d’amplitude aux petites valeurs de fréquence est une droite avec une pente nulle, et la phase tend vers zéro.
À la fréquence d’angle, l’amplitude asymptotique s’écarte de la ligne de pente zéro et la phase approche 45 degrés.
À des fréquences plus élevées, le graphique de magnitude forme une ligne de +20 dB/décennie, et la phase est de 90 degrés.
Un pôle simple est l’inverse d’un zéro simple. Cela signifie que les diagrammes de Bode basés sur des pôles reflètent le tracé zéro simple, réfléchi autour de l’axe horizontal.
Considérons une fonction de transfert de pôle quadratique.
Aux fréquences basses, le gain et l’angle de phase sont proches de zéro.
À la fréquence d’angle, l’écart de magnitude asymptotique dépend du facteur d’amortissement et l’angle de phase est de près de -90 degrés.
À des fréquences plus élevées, le graphique de magnitude forme une ligne droite avec une pente de -40 dB/décennie et une phase de -180 degrés.
Pour plus d’un pôle quadratique, la pente de la droite et le déphasage sont multipliés par le nombre de pôles.

réponse en fréquence

Sous la forme standard, la fonction de transfert est représentée en gain constant, pôles/zéros à l'origine, pôles/zéros simples et pôles/zéros quadratiques ; chacun contribuant de manière unique à la réponse globale du système. Le terme représente la grandeur du zéro simple:
<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="height: 36px;">
Le tracé de l'amplitude de Bode reste plat aux basses fréquences (approchant 0 dB) et commence à augmenter à 20 dB/décennie après une fréquence spécifique connue sous le nom de fréquence de coin ou de coupure, ω1. Il s'agit de la fréquence à laquelle la pente du tracé de magnitude change et la réponse réelle commence à s'écarter de l'approximation en ligne droite. Cet écart est quantifié à 3 dB à ω=ω_1.
L'angle de phase ϕ, exprimé par:
<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="height: 60px;">
L'angle de phase commence à 0° et s'approche asymptotiquement de 90° à mesure que la fréquence augmente. Pour des fréquences bien inférieures à la fréquence angulaire (ω≪ω_1), le terme jω/ω_1 est très petit, donc l'amplitude est négligeable et la phase est essentiellement nulle. À mesure que la fréquence s'approche de ω_1, cela conduit à un point d'amplitude de -3 dB et à un angle de phase de 45°. Pour des fréquences bien supérieures à ω_1 (ω≫ω_1), la pente de l'amplitude passe à 20 dB/décennie et la phase se stabilise à 90°.
<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="height: 160px;">
Pôle quadratique/zéro:
La grandeur et l'angle de phase d'un pôle quadratique sont:
<img alt="Equation 3" src="/files/ftp_upload/15142/15142_Equation_3.svg" style="height: 42px;">
<img alt="Equation 4" src="/files/ftp_upload/15142/15142_Equation_4.svg" style="height: 60px;">
Le tracé d'amplitude pour un pôle quadratique comporte deux parties : une réponse plate en dessous de la fréquence naturelle ω_n et une pente de -40 dB/décennie au-dessus de ω_n, le pic réel du tracé variant avec le facteur d'amortissement ζ2. Le tracé de phase d'un pôle quadratique diminue linéairement avec une pente de -90° par décennie, commençant à un dixième de la fréquence naturelle et se terminant à dix fois celle-ci, influencée par le facteur d'amortissement ζ2.

Réponse en fréquence

Considérons la fonction de transfert dans sa forme standard, avec des pôles et des zéros
Pour une fonction de transfert avec un zéro simple, le gain d’amplitude à ...

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bode plot slope and phase changes

Bode plots show how a transfer function changes with frequency. The standard form includes constant gain, poles and zeros at the origin, simple poles and zeros, and quadratic poles and zeros. Each part shapes the system response in a different way.

A simple zero sets the magnitude trend first. The Bode magnitude plot stays flat at low frequencies, near 0 dB, and then rises at 20 dB per decade after the corner or break frequency, ω1. That corner frequency is where the straight-line approximation starts to differ from the actual curve, and the error is 3 dB at ω = ω1.

The phase angle for a simple zero also changes with frequency. It starts at 0° and rises toward 90° as frequency increases. When ω is much smaller than ω1, the term jω/ω1 is very small, so the magnitude is negligible and the phase is close to zero. At ω = ω1, the magnitude is at the -3 dB point and the phase is 45°. When ω is much larger than ω1, the magnitude slope is 20 dB per decade and the phase settles near 90°.

Quadratic poles have a different shape on Bode plots. Their magnitude is flat below the natural frequency, ωn, and drops at -40 dB per decade above ωn. The actual peak depends on the damping factor, ζ2. Their phase plot decreases linearly at -90° per decade from one-tenth of ωn to ten times ωn, and that shape is also affected by ζ2.

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Bode Plot Slope and Phase Changes

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