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

Si consideri la funzione di trasferimento nella sua forma standard, con poli e zeri
Per una funzione di trasferimento con uno zero semplice, il guadagno di grandezza a piccoli valori di frequenza è una retta con pendenza zero e la fase si avvicina allo zero.
Alla frequenza d'angolo, la magnitudine asintotica devia dalla linea di pendenza zero e la fase si avvicina a 45 gradi.
A frequenze più elevate, il grafico della magnitudine forma una linea di +20 dB/decade e la fase è di 90 gradi.
Un polo semplice è il reciproco di uno zero semplice. Ciò significa che i diagrammi di Bode basati su poli rispecchiano il semplice grafico zero, riflesso attorno all'asse orizzontale.
Si consideri una funzione di trasferimento dei poli quadratica.
A frequenze più basse, il guadagno e l'angolo di fase si avvicinano allo zero.
Alla frequenza d'angolo, la deviazione asintotica dell'ampiezza dipende dal fattore di smorzamento e l'angolo di fase è di quasi -90 gradi.
A frequenze più elevate, il grafico della magnitudine forma una linea retta con una pendenza di -40 dB/decade e una fase di -180 gradi.
Per più di un polo quadratico, la pendenza della retta e lo sfasamento vengono moltiplicati per il numero di poli.

risposta in frequenza

Nella forma standard, la funzione di trasferimento è mostrata in guadagno costante, poli/zeri all'origine, poli/zeri semplici e poli/zeri quadratici; ciascuno contribuisce in modo univoco alla risposta complessiva del sistema. Il termine rappresenta la grandezza dello zero semplice:

<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="height: 36px;">

Il grafico della magnitudo di Bode rimane piatto alle basse frequenze (avvicinandosi a 0 dB) e inizia a salire a 20 dB/decennio dopo una frequenza specifica nota come frequenza d'angolo o di rottura, ω_1. Questa è la frequenza alla quale la pendenza del grafico della magnitudo cambia e la risposta effettiva inizia a deviare dall'approssimazione in linea retta. Questa deviazione è quantificata come 3 dB a ω=ω_1.

L'angolo di fase ϕ, espresso come:

<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="height: 60px;">

L'angolo di fase inizia a 0° e si avvicina asintoticamente a 90° all'aumentare della frequenza. Per frequenze molto inferiori alla frequenza d'angolo (ω≪ω_1), il termine jω/ω_1 è molto piccolo, quindi l'ampiezza è trascurabile e la fase è essenzialmente zero. Quando la frequenza si avvicina a ω_1, si ottiene un punto di magnitudo di -3 dB e un angolo di fase di 45°. Per frequenze molto più alte di ω_1 (ω≫ω_1), la pendenza dell'ampiezza cambia a 20 dB/decade e la fase si stabilizza a 90°.

<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="height: 160px;">

Polo/zero quadratico:

L'ampiezza e l'angolo di fase di un polo quadratico sono:

<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;">

Il grafico dell'ampiezza per un polo quadratico è composto da due parti: una risposta piatta al di sotto della frequenza naturale ω_n e una pendenza di -40 dB/decdimento al di sopra di ω_n, con il picco del grafico effettivo che varia con il fattore di smorzamento ζ_2. Il diagramma di fase per un polo quadratico diminuisce linearmente con una pendenza di -90° per decade, iniziando da un decimo della frequenza naturale e terminando a dieci volte quella, influenzato dal fattore di smorzamento ζ_2.

Risposta in frequenza

Si consideri la funzione di trasferimento nella sua forma standard, con poli e zeri
Per una funzione di trasferimento con uno zero semplice, il guadagno di grandezza a piccoli ...

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Electrical Engineering

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Unit 1

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Frequency Response

KEY TERMS AND CONCEPTS

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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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