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

考虑以标准形式表示的传递函数，包含极点和零点
对于具有简单零点的传递函数，在低频区域，幅值增益是一条斜率为零的直线，且相位趋近于零。
在转角频率处，渐近幅值偏离零斜率线，相位趋近于45度。
在较高频率下，幅值图呈现为一条+20 dB/十倍频程的直线，相位为90度。
一个一阶极点是一阶零点的倒数。这意味着基于极点的伯德图会以水平轴为对称轴，将一阶零点的图形进行镜像反射。
考虑一个二次极点传递函数。
在较低频率下，增益和相位角趋近于零。
在转角频率处，渐近幅值偏差取决于阻尼因子，且相位角接近 -90 度。
在较高频率下，幅值图呈现一条斜率为 -40 dB/十倍频程、相位为 -180 度的直线。
对于多个二次极点，直线的斜率和相位偏移均乘以极点的数量。

频率响应

在标准形式中，传递函数以恒定增益、原点处的极点/零点、简单极点/零点和二次极点/零点表示；每个极点/零点都对系统的整体响应产生独特的影响。术语表示简单零点的幅度：
<img alt="Equation 1" src="/files/ftp_upload/15142/15142_Equation_1.svg" style="text-align: center; height: 36px;">
波德幅度图在低频（接近 0 dB）时是保持平坦的，并且还会在特定频率（称为转折频率或断点频率）ω_1 之后以 20 dB/decade 的速度开始上升。这是幅度图斜率发生变化的频率，实际响应将会开始偏离直线的近似值。当 ω=ω_1 时，这种偏差可以量化为 3 dB。 
相位角 ϕ 表示为：
<img alt="Equation 2" src="/files/ftp_upload/15142/15142_Equation_2.svg" style="text-align: center; height: 60px;">
相位角从 0° 开始，将会随着频率的增加逐渐接近于 90°。对于远低于转角频率 （ω ≪ ω_1 ）的频率，jω/ω_1 项非常小，因此幅度可以忽略不计，相位基本上为零。当频率接近 ω_1 时，幅度会变为 -3 dB，相位角为 45°。对于远高于 ω_1 (ω≫ω_1) 的频率，幅度的斜率则会变为 20 dB/decade，并将相位稳定在 90°。
<img alt="Figure 1" src="/files/ftp_upload/15142/15142_Figure_1.svg" style="text-align: center; height: 160px;">
二次极点/零点：
二次极点的幅度和相位角为：
<img alt="Equation 3" src="/files/ftp_upload/15142/15142_Equation_3.svg" style="text-align: center; height: 42px;">
<img alt="Equation 4" src="/files/ftp_upload/15142/15142_Equation_4.svg" style="text-align: center; height: 60px;">
二次极点的幅度图分为两部分：自然频率 ω_n 以下的平坦响应和 ω_n 以上 -40 dB/decade 的斜率，实际图的峰值随阻尼系数会随着 ζ_2 的变化而变化。受阻尼系数 ζ_2 的影响，二次极点的相位图以 -90° 每十倍频程的斜率呈线性下降，从固有频率的十分之一开始，到固有频率的十倍处结束。

在标准形式中，传递函数表示为常数增益、位于原点的零极点、一阶零极点以及二阶零极点；每一项……

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Frequency Response

KEY TERMS AND CONCEPTS

关键术语与概念

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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electrical engineering

Bode Plot Slope and Phase Changes

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