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

Rozważmy funkcję przenoszenia w jej standardowej formie, z biegunami i zerami
W przypadku funkcji przenoszenia z prostym zerem, wzmocnienie wielkości przy małych wartościach częstotliwości jest linią prostą o zerowym nachyleniu, a faza zbliża się do zera.
Przy częstotliwości narożnej wielkość asymptotyczna odchyla się od linii zerowego nachylenia, a faza zbliża się do 45 stopni.
Przy wyższych częstotliwościach wykres wielkości tworzy linię +20 dB/dekadę, a faza wynosi 90 stopni.
Biegun prosty jest odwrotnością prostego zera. Oznacza to, że wykresy Bodego oparte na biegunach odzwierciedlają prosty wykres zerowy, odbity wokół osi poziomej.
Rozważmy kwadratową funkcję przenoszenia biegunów.
Przy niższych częstotliwościach wzmocnienie i kąt fazowy zbliżają się do zera.
Przy częstotliwości narożnej odchylenie asymptotyczne wielkości zależy od współczynnika tłumienia, a kąt fazowy wynosi prawie -90 stopni.
Przy wyższych częstotliwościach wykres wielkości tworzy linię prostą o nachyleniu -40 dB/dekadę i fazie -180 stopni.
W przypadku więcej niż jednego bieguna kwadratowego nachylenie linii i przesunięcie fazowe są mnożone przez liczbę biegunów.

odpowiedź na częstotliwość

W standardowej formie funkcja przenoszenia jest pokazana jako stałe wzmocnienie, bieguny/zera na początku, proste bieguny/zera i bieguny/zera kwadratowe; każdy z nich w unikalny sposób przyczynia się do ogólnej reakcji systemu. Termin reprezentuje wielkość prostego zera:

<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="height: 36px;">

Wykres wielkości Bodego pozostaje płaski przy niskich częstotliwościach (zbliżających się do 0 dB) i zaczyna rosnąć przy 20 dB/dekadę po określonej częstotliwości zwanej częstotliwością narożną lub załamania, ω_1. Jest to częstotliwość, przy której nachylenie wykresu wielkości zmienia się i rzeczywista odpowiedź zaczyna odbiegać od przybliżenia liniowego. Odchylenie to określa się ilościowo jako 3 dB przy ω=ω_1.

Kąt fazowy ϕ, wyrażony jako:

<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="height: 60px;">

Kąt fazowy zaczyna się od 0° i asymptotycznie zbliża się do 90° wraz ze wzrostem częstotliwości. Dla częstotliwości znacznie niższych niż częstotliwość narożna (ω≪ω_1) człon jω/ω_1 jest bardzo mały, więc wielkość jest pomijalna, a faza wynosi zasadniczo zero. Gdy częstotliwość zbliża się do ω_1, co prowadzi do punktu wielkości -3 dB i kąta fazowego 45°. Dla częstotliwości znacznie wyższych niż ω_1 (ω≫ω_1) nachylenie wielkości zmienia się do 20 dB/dekadę, a faza ustala się na poziomie 90°.

<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="height: 160px;">

Biegun kwadratowy/zero:

Wielkość i kąt fazowy bieguna kwadratowego wynosi:

<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;">

Wykres amplitudy dla bieguna kwadratowego składa się z dwóch części: płaskiej odpowiedzi poniżej częstotliwości własnej ω_n i nachylenia -40 dB/dekadę powyżej ω_n, przy czym rzeczywisty szczyt wykresu zmienia się wraz ze współczynnikiem tłumienia ζ_2. Wykres fazowy dla bieguna kwadratowego maleje liniowo z nachyleniem -90° na dekadę, zaczynając od jednej dziesiątej częstotliwości własnej i kończąc na dziesięciokrotności częstotliwości własnej, na co wpływa współczynnik tłumienia ζ_2.

Odpowiedź na częstotliwość

Rozważmy funkcję przenoszenia w jej standardowej formie, z biegunami i zerami
W przypadku funkcji przenoszenia z prostym zerem, przyrost wielkości przy małym ...

Education

JoVE Core

Electrical Engineering

Elektrotechnika

Unit 1

Jednostka 1

Frequency Response

Charakterystyka częstotliwościowa

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Transfer function and Bode Plots-II

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.

Bode Plot Slope and Phase Changes

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transfer function and bode plots-ii

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...

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electrical engineering

Bode Plot Slope and Phase Changes

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