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

Weźmy pod uwagę funkcję przenoszenia systemu wyrażoną w formie standardowej.
Funkcja przenoszenia ma stałe wzmocnienie, zero i biegun w punkcie początkowym, proste zero, prosty biegun, biegun kwadratowy i zero kwadratowe.
Wykresy Bodego dla poszczególnych czynników są wykreślane oddzielnie i łączone graficznie.
Dla dodatniego składnika stałego wzmocnienia wzmocnienie wielkości jest stałe, a kąt fazowy jest zerowy i niezależny od częstotliwości.
W przypadku ujemnego składnika stałego wzmocnienia wzmocnienie wielkości pozostaje takie samo, a kąt fazowy wynosi ±180°.
W przypadku funkcji przenoszenia z zerem w punkcie początkowym wykres wielkości reprezentuje linię prostą o dodatnim nachyleniu 20 dB/dekadę. Wykres fazowy przedstawia stały kąt fazowy wynoszący 90°.
W przypadku bieguna w punkcie początkowym nachylenie wykresu wielkości wynosi -20 dB/dekadę, a przesunięcie fazowe wynosi -90°.
Jeśli w punkcie początkowym układu współrzędnych znajduje się więcej niż jeden biegun lub zero, nachylenie wykresu wielkości i przesunięcie fazowe mnoży się przez liczbę biegunów lub zer obecnych w punkcie początkowym.

odpowiedź na częstotliwość

Funkcja transferu przedstawiona w swojej standardowej postaci integruje stałe wzmocnienie elementów, zera i bieguny w początku, proste zera i bieguny oraz kwadratowe bieguny i zera. Funkcję przenoszenia można zapisać jako H(ω):

<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="height: 40px;">

Funkcję przenoszenia, H(ω), często wyrażaną w postaci standardowej, wyprowadza się poprzez normalizację współczynników wielomianu funkcji przenoszenia. Bieguny (jω) i zera (jω) to częstotliwości krytyczne, w przypadku których wielkość i faza sygnału wyjściowego systemu ulegają znaczącym zmianom.

Funkcja przenoszenia ma stały człon K, o wielkości 20 log_10K i kącie fazowym 0° dla dodatniej wartości K. Zarówno wielkość, jak i faza są stałe z częstotliwością. Jeśli K jest ujemne, wielkość pozostaje niezmieniona, ale faza wynosi ± 180 °. Gdy wzmocnienie K = 1, wielkość wyniesie zero wraz z kątem fazowym. Na wykresie Bodego wielkość wyraża się w decybelach (dB) jako 20log_10K, a jej faza pozostaje na poziomie 0∘ lub 180∘, w zależności od znaku K.

Biegun / zero na początku:

Biegun lub zero na początku ma decydujący wpływ na wykres. Zero (jω)^+1 na początku ma wielkość 20 log_10 ω, a faza wynosi 90°. Nachylenie wykresu wielkości wynosi 20 dB/dekadę, a faza jest stała wraz z częstotliwością.

Dla bieguna (jω)^-1 w punkcie początkowym wielkość wynosi -20dB/dekadę, a faza -90°. Ogólnie rzecz biorąc, dla (jω)^N, gdzie N jest liczbą całkowitą, wykres wielkości będzie miał nachylenie -20NdB/dekadę, podczas gdy faza wynosi 90N°.

Stała wielkość K, bieguny/zera na początku mają kąt fazowy, który nie zmienia się wraz z częstotliwością. W przypadku zer/biegunów kąt fazowy zmienia się, gdy zmienia się liczba biegunów/zer na początku.

Odpowiedź na częstotliwość

Weźmy pod uwagę funkcję przenoszenia systemu wyrażoną w formie standardowej.
Funkcja przenoszenia ma stałe wzmocnienie, zero i biegun na początku, ...

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-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.

Transfer Function Gain and Origin Effects

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transfer function and bode plots-i

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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.

Transfer Function Gain and Origin Effects | JoVE K12

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electrical engineering

Transfer Function Gain and Origin Effects

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