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

考虑以标准形式表示的系统传递函数。
该传递函数具有恒定增益、一个位于原点的零点和极点、一个简单零点、一个简单极点、一个二次极点以及一个二次零点。
各个因子的伯德图分别绘制，然后通过图形方式合并。
对于正的常数增益项，其幅值增益保持恒定，相位角为零且不随频率变化。
对于负的常数增益项，其幅值增益保持不变，相位角为 ±180°。
对于在原点处有零点的传递函数，其幅值图表现为一条斜率为正的直线，斜率为 20 dB/decade。相位图则显示一个恒定的相位角，为 90°。
对于位于原点的极点，幅值图的斜率为 -20 dB/decade，相位偏移为 -90°。
如果在原点处存在多个极点或零点，则幅频特性曲线的斜率和相位偏移将乘以原点处极点或零点的数量。

频率响应

以标准形式呈现的传递函数整合了元素的恒定增益、原点处的零点和极点、简单的零点和极点以及二次极点和零点。传递函数可以用 H(ω) 来进行表示：
<img alt="Equation 1" src="/files/ftp_upload/15141/15141_Equation_1.svg" style="text-align: center; height: 40px;">
以标准形式来进行表示的传递函数 H(ω) 是通过对传递函数的多项式系数进行归一化而得出的。极点（jω）和零点（jω）是系统输出的幅度和相位发生显著变化的临界频率。
增益 K：
传递函数中有一个常数项 K，当 K 为正时，其幅度为 20 log_10 K，相位角为 0°。幅度和相位都与频率成正比。如果 K 为负，幅度则保持不变，但相位为 ±180°。当增益 K = 1 时，幅度和相位角均为零。在波德图中，可以将幅度用分贝（dB）表示为 20 log_10 K，其相位保持在 0° 或 180°，这具体取决于 K 的符号。
原点的极点/零点：
原点上的极点或零点对其中的曲线有着决定性的影响。位于原点处的零点 (jω)^+1 的幅度为 20 log_10 ω，相位为 90°。幅度图的斜率为 20 dB/decade，相位会随着频率的变化而变化。
对于原点的极点 (jω)^-1 来说，幅度为 -20 dB/decade，相位为 -90°。通常，对于 (jω)^N 来说（其中 N 为整数），幅度图的斜率为 -20N dB/decade，相位为 (90N)°。
恒定幅度 K，位于原点处的极点/零点具有不随频率变化的相位角。对于零点/极点来说，当原点处极点/零点的数量发生变化时，相位角也会随之发生变化。

考虑以标准形式表示的系统的传递函数。  
该传递函数具有一个恒定增益、一个零点以及位于原点的一个极点，一个……

Education

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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-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.

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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

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

Transfer Function Gain and Origin Effects

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