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

Рассмотрим передаточную функцию в ее стандартном виде, с полюсами и нулями
Для передаточной функции с простым нулем усиление амплитуды при малых значениях частоты представляет собой прямую линию с нулевым наклоном, а фаза стремится к нулю.
На угловой частоте асимптотическая величина отклоняется от нулевой линии наклона, и фаза приближается к 45 градусам.
На более высоких частотах график амплитуды образует линию +20 дБ/декаду, а фаза равна 90 градусам.
Простой полюс — это величина, обратная простому нулю. Это означает, что столбовые графики Боде отражают простой нулевой график, отраженный вокруг горизонтальной оси.
Рассмотрим квадратичную функцию переноса полюсов.
На более низких частотах коэффициент усиления и фазовый угол приближаются к нулю.
На угловой частоте асимптотическое отклонение амптотической величины зависит от коэффициента демпфирования, а фазовый угол составляет почти -90 градусов.
На более высоких частотах график амплитуды образует прямую линию с наклоном -40 дБ/декаду и фазой -180 градусов.
Для более чем одного квадратичного полюса наклон линии и фазовый сдвиг умножаются на число полюсов.

Частотная характеристика

В стандартной форме передаточная функция отображается в виде постоянного усиления, полюсов/нулей в начале координат, простых полюсов/нулей и квадратичных полюсов/нулей; каждый из которых вносит свой уникальный вклад в общую реакцию системы. Этот термин представляет собой величину простого нуля:
<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="height: 36px;">
График Боде остается плоским на низких частотах (приближаясь к 0 дБ) и начинает возрастать на 20 дБ/декаду после определенной частоты, известной как угловая частота или частота излома, ω_1. Это частота, на которой наклон графика величины меняется и фактический отклик начинает отклоняться от прямолинейного приближения. Это отклонение количественно оценивается как 3 дБ при ω=ω_1.
Фазовый угол φ, выражаемый как:
<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="height: 60px;">
Фазовый угол начинается с 0° и асимптотически приближается к 90° по мере увеличения частоты. Для частот, намного меньших, чем угловая частота (ω≪ω_1), член jω/ω_1 очень мал, поэтому величина незначительна, а фаза практически равна нулю. Когда частота приближается к ω_1, это приводит к значению -3 дБ величины, и фазовому углу 45°. Для частот, намного превышающих ω1 (ω≫ω_1), наклон изменяется до 20 дБ/декаду, а фаза устанавливается на уровне 90°.
<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="height: 160px;">
Квадратичный полюс/ноль:
Величина и фазовый угол квадратичного полюса равны:
<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;">
График величины для квадратичного полюса состоит из двух частей: плоский отклик ниже собственной частоты ω_n и наклон -40 дБ/декаду выше ω_n, причем фактический пик графика изменяется в зависимости от коэффициента затухания ζ_2. Фазовый график для квадратичного полюса уменьшается линейно с наклоном -90° за декаду, начиная с одной десятой собственной частоты и заканчивая десятикратной частотой под влиянием коэффициента затухания ζ_2.

Рассмотрим передаточную функцию в ее стандартном виде, с полюсами и нулями
Для передаточной функции с простым нулем усиление амплитуды при малом ...

Education

JoVE Core

Electrical Engineering

Электротехника

Unit 1

Раздел 1

Frequency Response

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

network function of a circuit

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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...

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...

bode plots construction

Reading Frequency Response with Bode Plots

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Series Resonance in RLC Circuits

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characteristics-of-series-resonant-circuit

Series Resonance and RLC Filter Behavior

Series resonance in an RLC circuit happens when the inductor, capacitor, and resistor are connected in series. At the resonant frequency, the inductive reactance and capacitive reactance are equal in size but opposite in sign, so they cancel each other. As a result, the circuit impedance becomes very small and is mainly set by the resistance. The resonant frequency of a series RLC circuit can be calculated from its inductance and capacitance. At this frequency, the current reaches its maximum...

parallel resonance

Parallel RLC Tuning at Resonance

A parallel RLC circuit uses a resistor, inductor, and capacitor connected to the same nodes. Because they share the same voltage, the circuit is analyzed with admittance, which describes how easily current can flow. In a parallel RLC circuit, admittance helps show how the circuit responds as the frequency changes. Resonance happens when the net reactance is zero. At that point, the capacitive and inductive effects cancel each other out. The resonant frequency is found from this condition. At...

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Op Amp Gain and Bode Plot Basics

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Audio Frequency Control with Passive Filters

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key terms and concepts

Learn how simple zeros and quadratic poles shape Bode plot magnitude and phase, including corner frequency, 3 dB shift, and damping effects.

Bode Plot Slope and Phase Changes | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Bode Plot Slope and Phase Changes

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