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

Betrachten Sie die Übertragungsfunktion in ihrer Standardform mit Polen und Nullen
Bei einer Übertragungsfunktion mit einer einfachen Null ist der Betragsgewinn bei kleinen Frequenzwerten eine gerade Linie mit einer Steigung von Null, und die Phase nähert sich Null.
Bei der Eckfrequenz weicht die asymptotische Größe von der Null-Steigungslinie ab, und die Phase nähert sich 45 Grad.
Bei höheren Frequenzen bildet das Magnitudendiagramm eine Linie von +20 dB/Dekade, und die Phase beträgt 90 Grad.
Ein einfacher Pol ist der Kehrwert einer einfachen Null. Das bedeutet, dass polbasierte Bode-Diagramme das einfache Nulldiagramm widerspiegeln, das um die horizontale Achse reflektiert wird.
Betrachten Sie eine quadratische Polübertragungsfunktion.
Bei niedrigeren Frequenzen gehen die Verstärkung und der Phasenwinkel gegen Null.
Bei der Eckfrequenz hängt die asymptotische Größenabweichung vom Dämpfungsfaktor ab, und der Phasenwinkel beträgt nahezu -90 Grad.
Bei höheren Frequenzen bildet das Magnitudendiagramm eine gerade Linie mit einer Steigung von -40 dB/Dekade und einer Phase von -180 Grad.
Bei mehr als einem quadratischen Pol werden die Steigung der Linie und die Phasenverschiebung mit der Anzahl der Pole multipliziert.

frequenzgang

In der Standardform wird die Übertragungsfunktion mit konstanter Verstärkung, Polen/Nullen am Ursprung, einfachen Polen/Nullen und quadratischen Polen/Nullen dargestellt; jede davon trägt auf einzigartige Weise zur Gesamtantwort des Systems bei. Der Term stellt die Größe der einfachen Null dar:

<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="text-align: center; height: 36px;">

Das Bode-Größendiagramm bleibt bei niedrigen Frequenzen flach (nähert sich 0 dB) und beginnt nach einer bestimmten Frequenz, die als Eck- oder Knickfrequenz ω_1 bezeichnet wird, mit 20 dB/Dekade anzusteigen. Dies ist die Frequenz, bei der sich die Steigung des Größendiagramms ändert und die tatsächliche Antwort von der geradlinigen Näherung abzuweichen beginnt. Diese Abweichung wird bei ω=ω_1 mit 3 dB quantifiziert.

Der Phasenwinkel ϕ, ausgedrückt als:

<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="text-align: center; height: 60px;">

Der Phasenwinkel beginnt bei 0° und nähert sich asymptotisch 90°, wenn die Frequenz zunimmt. Bei Frequenzen, die deutlich niedriger als die Eckfrequenz (ω≪ω_1) sind, ist der Term jω/ω_1 sehr klein, sodass die Größe vernachlässigbar ist und die Phase im Wesentlichen Null beträgt. Wenn sich die Frequenz ω_1 nähert, führt dies zu einem -3 dB-Punkt in der Größe und einem Phasenwinkel von 45°. Bei Frequenzen, die deutlich höher als ω_1 sind (ω≫ω_1), ändert sich die Steigung der Größe auf 20 dB/Dekade und die Phase stabilisiert sich bei 90°.

<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="text-align: center; height: 160px;">

Quadratischer Pol/Nullpunkt:

Größe und Phasenwinkel eines quadratischen Pols sind:

<img alt="Equation 3" src="/files/ftp_upload/15142/15142_Equation_3.svg" style="text-align: center; height: 42px;">

<img alt="Equation 4" src="/files/ftp_upload/15142/15142_Equation_4.svg" style="text-align: center; height: 60px;">

Das Amplitudendiagramm für einen quadratischen Pol besteht aus zwei Teilen: einer flachen Reaktion unterhalb der Eigenfrequenz ω_n und einer Steigung von -40 dB/Dekade oberhalb ω_n, wobei die Spitze des tatsächlichen Diagramms mit dem Dämpfungsfaktor ζ_2 variiert. Das Phasendiagramm für einen quadratischen Pol nimmt linear mit einer Steigung von -90° pro Dekade ab, beginnend bei einem Zehntel der Eigenfrequenz und endend bei dem Zehnfachen davon, beeinflusst durch den Dämpfungsfaktor ζ_2.

Frequenzgang

Betrachten Sie die Übertragungsfunktion in ihrer Standardform mit Polen und Nullen
Bei einer Übertragungsfunktion mit einer einfachen Null wird der Magnitudengewinn bei kleinen ...

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Unit 1

Einheit 1

Frequency Response

KEY TERMS AND CONCEPTS

ZENTRALE BEGRIFFE UND KONZEPTE

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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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