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A Description of Negative Frequency and Phase Angle by Rotating Vector
DOI: 10.12677/OJCS.2020.93007, PDF, HTML, XML, 下载: 359  浏览: 1,160  科研立项经费支持

Abstract: In some teaching materials of signals and systems, negative frequency is said to have no physical meaning, and the definition of phase angle is not strict enough. In this paper, the negative fre-quency and phase angle are described by the method of rotation vector. From the geometric sense, if the positive frequency is the counter clockwise rotation angular velocity of the vector, then the negative frequency is the angular velocity of clockwise rotation. From the engineering point of view, it can correspond to the reverse rotation of the generator. Therefore, negative frequency not only has clear physical significance, but also has important engineering application values. In general, the phase angle is defined by the arctangent trigonometric function, but the range of the phase angle is −π/2 to π/2, the actual phase angle should be −π to π. This problem can be strictly defined by rotation vector.

1. 引言

2. 傅里叶级数的描述

2.1. 三角形式的傅里叶级数

$\begin{array}{l}x\left(t\right)={a}_{0}+\underset{n=1}{\overset{\infty }{\sum }}{a}_{n}\mathrm{cos}\left(n{\omega }_{0}t\right)+{b}_{n}\mathrm{sin}\left(n{\omega }_{0}t\right)\\ {a}_{0}=\frac{1}{T}{\int }_{{t}_{0}}^{{t}_{0}+T}x\left(t\right)\text{d}t\\ {a}_{n}=\frac{2}{T}{\int }_{{t}_{0}}^{{t}_{0}+T}x\left(t\right)\mathrm{cos}\left(n{\omega }_{0}t\right)\text{d}t\\ {b}_{n}=\frac{2}{T}{\int }_{{t}_{0}}^{{t}_{0}+T}x\left(t\right)\mathrm{sin}\left(n{\omega }_{0}t\right)\text{d}t\end{array}$ (1)

$\begin{array}{l}x\left(t\right)={a}_{0}+\underset{n=1}{\overset{\infty }{\sum }}{A}_{n}\left[\mathrm{cos}\left(n{\omega }_{0}t+{\phi }_{n}\right)\right]\\ {A}_{n}=\sqrt{{a}_{n}^{2}+{b}_{n}^{2}}\\ {\phi }_{n}=-\mathrm{arctan}\left({b}_{n}/{a}_{n}\right)\end{array}$ (2)

2.2. 指数形式的傅里叶级数

$\begin{array}{l}x\left(t\right)=\underset{n=-\infty }{\overset{\infty }{\sum }}{c}_{n}{\text{e}}^{jn{\omega }_{0}t}\\ {c}_{n}=\frac{1}{T}{\int }_{{t}_{0}}^{{t}_{0}+T}x\left(t\right){\text{e}}^{-jn{\omega }_{0}t}\text{d}t=\frac{1}{2}{\stackrel{˙}{A}}_{n}{\text{e}}^{j{\phi }_{n}}=\frac{1}{2}\left({a}_{n}-j{b}_{n}\right)\end{array}$ (3)

3. 旋转矢量的定义

Figure 1. Rotation vector

Figure 2. Rotation vector and complex sine wave

$r{\text{e}}^{j\left(\omega t+\theta \right)}=r\left[\mathrm{cos}\left(\omega t+\theta \right)+j\mathrm{sin}\left(\omega t+\theta \right)\right]$ (4)

$r{\text{e}}^{j\left(\omega t+\theta \right)}$ 是简单的复正弦波，其立体图不易表达，所以一般用两张图来表示，如图3所示。

Figure 3. Real part and imaginary part of complex sine wave

Figure 4. The superposition of two conjugate complex sinusoids forms a real sine wave

$\mathrm{cos}\left(\omega t\right)=\frac{{\text{e}}^{j\omega t}+{\text{e}}^{-j\omega t}}{2}$, $\mathrm{sin}\left(\omega t\right)=\frac{{\text{e}}^{j\omega t}-{\text{e}}^{-j\omega t}}{2j}$

4. 用旋转矢量解释傅里叶级数中的负频率

Figure 5. The rotation vector is used to express the parameters of Fourier series

$v=\frac{\text{d}s}{\text{d}t}$, $\omega =\frac{\text{d}\theta }{\text{d}t}$

5. 用旋转矢量定义相位角

${\phi }_{n}=-\mathrm{arctan}\left(\frac{{b}_{n}}{{a}_{n}}\right)$$\mathrm{tan}{\phi }_{n}=-\frac{{b}_{n}}{{a}_{n}}$ (5)

$\mathrm{cos}\left({\omega }_{0}t\right)+\mathrm{sin}\left({\omega }_{0}t\right)=\sqrt{2}\mathrm{cos}\left({\omega }_{0}t+\frac{\pi }{4}\right)$

$-\mathrm{cos}\left({\omega }_{0}t\right)-\mathrm{sin}\left({\omega }_{0}t\right)=\sqrt{\text{2}}\mathrm{cos}\left({\omega }_{0}t+\frac{5\pi }{4}\right)$ (6)

Figure 6. Relationship between phase angle φn and an, bn

A： ${a}_{n}>0$${b}_{n}=0$${\phi }_{n}=0$ B： ${a}_{n}>0$${b}_{n}>0$$0<{\phi }_{n}<\pi /2$

C： ${a}_{n}=0$${b}_{n}>0$${\phi }_{n}=\pi /2$ D： ${a}_{n}<0$${b}_{n}>0$$\pi /2<{\phi }_{n}<\pi$

E： ${a}_{n}<0$${b}_{n}=0$${\phi }_{n}=\pi$ F： ${a}_{n}<0$${b}_{n}<0$$-\pi <{\phi }_{n}<-\pi /2$

G： ${a}_{n}=0$${b}_{n}<0$${\phi }_{n}=-\pi /2$ H： ${a}_{n}>0$${b}_{n}<0$$-\pi /2<{\phi }_{n}<0$

O： ${a}_{n}=0$${b}_{n}=0$，φn取任意值

${\phi }_{n}=\mathrm{arg}\left({a}_{n},{b}_{n}\right)$ (7)

${\phi }_{n}=\mathrm{arg}\left({c}_{n}\right)$ (8)

6. 结论

NOTES

*通讯作者。

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