Name: transfer function gain and origin effects
Uploaded: 2024-07-08T13:20:02.000Z
Duration: 1 min 19 s
Channel: JoVE

Betrachten wir die Übertragungsfunktion eines Systems, das in Standardform ausgedrückt wird.
Die Übertragungsfunktion hat eine konstante Verstärkung, eine Null und einen Pol am Ursprung, eine einfache Null, einen einfachen Pol, einen quadratischen Pol und eine quadratische Null.
Bode-Plots für einzelne Faktoren werden separat dargestellt und grafisch kombiniert.
Für den positiven konstanten Verstärkungsterm ist die Magnitudenverstärkung konstant und der Phasenwinkel ist null und frequenzunabhängig.
Für den negativen Term der konstanten Verstärkung bleibt die Magnitudenverstärkung gleich und der Phasenwinkel beträgt ±180°.
Für eine Übertragungsfunktion mit einer Null am Ursprung stellt das Magnitudendiagramm eine gerade Linie mit einer positiven Steigung von 20 dB/Dekade dar. Das Phasendiagramm bildet einen konstanten Phasenwinkel von 90° ab.
Für einen Pol am Ursprung beträgt die Steigung des Magnitudendiagramms -20 dB/Dekade und die Phasenverschiebung -90°.
Wenn es mehr als einen Pol oder eine Null am Ursprung gibt, werden die Steigung des Magnitudendiagramms und die Phasenverschiebung mit der Anzahl der Pole oder Nullstellen multipliziert, die am Ursprung vorhanden sind.

frequenzgang

Eine Übertragungsfunktion in ihrer Standardform integriert die konstante Verstärkung der Elemente, die Nullstellen und Pole am Ursprung, einfache Nullstellen und Pole sowie quadratische Pole und Nullstellen. Die Übertragungsfunktion kann als H(ω) geschrieben werden:

<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="text-align: center; height: 40px;">

Die Übertragungsfunktion H(ω), die oft in der Standardform ausgedrückt wird, wird durch Normalisierung der Polynomkoeffizienten der Übertragungsfunktion abgeleitet. Die Pole (jω) und Nullstellen (jω) sind kritische Frequenzen, bei denen sich Betrag und Phase des Systemausgangs erheblich ändern.

Verstärkung, K:

Die Übertragungsfunktion hat einen konstanten Term K mit einem Betrag von 20 log_10K und einem Phasenwinkel von 0° für einen positiven Wert von K. Sowohl Betrag als auch Phase sind mit der Frequenz konstant. Wenn K negativ ist, bleibt der Betrag unverändert, aber die Phase beträgt ±180°. Wenn die Verstärkung K =1 ist, wird der Betrag zusammen mit dem Phasenwinkel Null. In Bode-Diagrammen wird die Magnitude in Dezibel (dB) als 20log_10K ausgedrückt, und ihre Phase bleibt bei 0∘ oder 180∘, je nach Vorzeichen von K.

Pol/Null am Ursprung:

Ein Pol oder eine Null am Ursprung hat einen entscheidenden Einfluss auf das Diagramm. Eine Null (jω)^+1 am Ursprung hat eine Magnitude von 20 log_10 ω und die Phase beträgt 90°. Die Steigung des Magnitudendiagramms beträgt 20 dB/Dekade, und die Phase ist mit der Frequenz konstant.

Für einen Pol (jω)^-1 am Ursprung beträgt die Magnitude -20 dB/Dekade und die Phase -90°. Im Allgemeinen hat das Magnitudendiagramm für (jω)^N, wobei N eine Ganzzahl ist, eine Steigung von -20 N dB/Dekade, während die Phase 90 N° beträgt.

Bei der konstanten Magnitude K haben Pole/Nullen an den Ursprüngen einen Phasenwinkel, der sich mit der Frequenz nicht ändert. Bei Nullstellen/Polen ändert sich der Phasenwinkel, wenn sich die Anzahl der Pole/Nullstellen im Ursprung ändert.

Frequenzgang

Betrachten wir die Übertragungsfunktion eines Systems, das in Standardform ausgedrückt wird.
Die Übertragungsfunktion hat eine konstante Verstärkung, eine Null und einen Pol am Ursprung, eine ...

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Frequency Response

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Transfer function and Bode Plots-I

transfer function gain and origin effects

Transfer function gain and poles or zeros at the origin shape a system's Bode plot. The transfer function is often written in standard form after the polynomial coefficients are normalized. In that form, the constant gain, zeros, poles, and quadratic terms all appear clearly. The poles and zeros mark critical frequencies where the output magnitude and phase change noticeably.

The constant gain K creates a fixed magnitude and phase term. Its magnitude is 20 log 10 K in decibels, and its phase is 0° when K is positive. If K is negative, the magnitude stays the same, but the phase becomes ±180°. When K = 1, the magnitude is 0 dB and the phase angle is 0°.

A zero at the origin has a strong effect on the Bode plot. For a zero at the origin, the magnitude is 20 log 10 ω, and the phase is 90°. The magnitude rises with a slope of 20 dB/decade, and the phase stays constant as frequency changes. A pole at the origin acts in the opposite way. Its magnitude slope is -20 dB/decade, and its phase is -90°.

More generally, a factor of (jω)^N changes the plot by an integer amount. For a zero at the origin, the magnitude slope becomes 20N dB/decade and the phase is 90N°. For a pole at the origin, the magnitude slope becomes -20N dB/decade, and the phase is also 90N°. In both cases, the constant gain K and the poles or zeros at the origin keep a phase angle that does not change with frequency. The phase changes when the number of poles or zeros at the origin changes.

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Explore how constant gain, poles, and zeros at the origin affect transfer functions and Bode plots, including dB slope and phase changes.

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electrical engineering

Transfer Function Gain and Origin Effects

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