Clark变换

作用

参考：

https://blog.csdn.net/chenjianbo88/article/details/53027298

https://zhuanlan.zhihu.com/p/147659820

preview

将平面三相坐标系 $I_aI_bI_c$ 变换为平面直角坐标系 $I_\alpha$ $I_\beta$
在这里插入图片描述
写成矩阵：

Python模拟

sin函数：https://blog.csdn.net/henni_719/article/details/77367294

正弦波信号：https://blog.csdn.net/weixin_39124421/article/details/103817951

np.linspace():https://blog.csdn.net/Asher117/article/details/87855493

import numpy as np
import matplotlib.pyplot as plt


def clark_transformation(input_matrix):
    trans_arr = np.mat([[1, -1 / 2, -1 / 2], [0, np.sqrt(3) / 2, -np.sqrt(3) / 2]])
    return trans_arr * input_matrix


def show_plot():
    # init
    plt.figure(figsize=(50,10))

    # process
    lin = np.linspace(0, 50, num=1000)
    print(lin)
    sin_wave1 = np.sin(lin)
    phi = np.pi * 2 / 3
    sin_wave2 = np.sin(lin - phi)
    sin_wave3 = np.sin(lin - phi * 2)

    # draw
    plt.subplot(211)
    plt.grid()
    plt.title('Source')
    plt.plot(lin, sin_wave1)
    plt.plot(lin, sin_wave2)
    plt.plot(lin, sin_wave3)

    # clark transformation
    trans_wave1 = np.empty(lin.shape)
    trans_wave2 = np.empty(lin.shape)
    for i in range(len(lin)):
        input_array = np.mat([[sin_wave1[i]], [sin_wave2[i]], [sin_wave3[i]]])
        output_array = clark_transformation(input_array)
        trans_wave1[i] = output_array[0]
        trans_wave2[i] = output_array[1]

    # draw
    plt.subplot(212)
    plt.grid()
    plt.title('Trans')
    plt.plot(lin, trans_wave1)
    plt.plot(lin, trans_wave2)

    plt.show()


show_plot()

效果：

在这里插入图片描述

上图为相位相差 $\frac{2\pi}{3}$ 的三个正弦波

下图为经过Clark变换得到的两个正弦波

Park变换

作用

让新的坐标系跟着theta角旋转，使得投影到该坐标系上的值为定值，达到让非线性线性化的效果
在这里插入图片描述

写成矩阵：（其实就是一个旋转矩阵）
在这里插入图片描述

Python模拟

import numpy as np
import matplotlib.pyplot as plt


def clark_transformation(input_matrix):
    trans_arr = np.mat([[1, -1 / 2, -1 / 2], [0, np.sqrt(3) / 2, -np.sqrt(3) / 2]])
    return trans_arr * input_matrix


def park_transformation(input_matrix, theta):
    trans_arr = np.mat([[np.cos(theta), np.sin(theta)], [-np.sin(theta), np.cos(theta)]])
    return trans_arr * input_matrix


def show_plot():
    # init
    plt.figure(figsize=(20, 10))

    # process
    lin = np.linspace(0, 50, num=1000)
    print(lin)
    sin_wave1 = np.sin(lin)
    phi = np.pi * 2 / 3
    sin_wave2 = np.sin(lin - phi)
    sin_wave3 = np.sin(lin - phi * 2)

    # draw1
    plt.subplot(311)
    plt.grid()
    plt.title('Source')
    plt.plot(lin, sin_wave1)
    plt.plot(lin, sin_wave2)
    plt.plot(lin, sin_wave3)

    # clark transformation
    clark_trans_wave1 = np.empty(lin.shape)
    clark_trans_wave2 = np.empty(lin.shape)
    for i in range(len(lin)):  # i means index, lin[i] means theta
        input_array = np.mat([[sin_wave1[i]],
                              [sin_wave2[i]],
                              [sin_wave3[i]]])
        output_array = clark_transformation(input_array)
        clark_trans_wave1[i] = output_array[0]
        clark_trans_wave2[i] = output_array[1]

    # draw2
    plt.subplot(312)
    plt.grid()
    plt.title('Clark')
    plt.plot(lin, clark_trans_wave1)
    plt.plot(lin, clark_trans_wave2)

    # park transformation
    park_trans_wave1 = np.empty(lin.shape)
    park_trans_wave2 = np.empty(lin.shape)
    for i in range(len(lin)):  # i means index, lin[i] means theta
        input_array = np.mat([[clark_trans_wave1[i]],
                              [clark_trans_wave2[i]]])
        output_array = park_transformation(input_array, lin[i])
        park_trans_wave1[i] = output_array[0]
        park_trans_wave2[i] = output_array[1]

    # draw3
    plt.subplot(313)
    plt.grid()
    plt.title('Park')
    plt.plot(lin, park_trans_wave1)
    plt.plot(lin, park_trans_wave2)

    # show plot
    plt.show()


show_plot()