利用Python制作动图演示坐标变换理论
利用Python制作动图演示坐标变换理论
永磁同步电机是一个非线性、强耦合的物理系统，因而不便直接进行控制。后有研究人员创造性的提出了坐标变换理论(后逐渐成为矢量控制的一个部分)，让永磁同步电机得以转化为一种类直流电机模型，此时电机成为了一个弱耦合系统，电压表达式中仍含有直流耦合项。本文通过Python演示了矢量控制中的坐标变换过程，读者需对矢量控制中的坐标变换理论较为了解。
电脑需要一个能运行Python代码的环境，我使用的软件环境
- Sublime Text
- 可能需要借助anaconda安装一些依赖库
坐标变换理论的一种数学形式
下文以三相电流的数学形式描述坐标变换理论，此处采用余弦形式
[ i a i b i c ] = [ I 1 cos ( ω e t ) I 1 cos ( ω e t − 2 3 π ) I 1 cos ( ω e t + 2 3 π ) ] \left[ \begin{array}{c} i_a\\ i_b\\ i_c\\ \end{array} \right] =\left[ \begin{array}{c} I_1\cos \left( \omega _et \right)\\ I_1\cos \left( \omega _et-\frac{2}{3}\pi \right)\\ I_1\cos \left( \omega _et+\frac{2}{3}\pi \right)\\ \end{array} \right] ⎣⎡iaibic⎦⎤=⎣⎡I1cos(ωet)I1cos(ωet−32π)I1cos(ωet+32π)⎦⎤
式中： I 1 I_1 I1是相电流幅值， ω e \omega_e ωe是电频率， i a i_a ia、 i b i_b ib、 i c i_c ic是相电流
三相自然坐标系至两相静止坐标系的变换矩阵为
T 3 s / 2 s = 2 3 [ 1 − 1 2 − 1 2 0 3 2 − 3 2 2 2 2 2 2 2 ] \boldsymbol{T}_{3s/2s}=\frac{2}{3}\left[ \begin{matrix} 1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}\\ \end{matrix} \right] T3s/2s=32⎣⎢⎡1022−212322−21−2322⎦⎥⎤
即
[ i α i β ] = T 3 s / 2 s [ i a i b i c ] \left[ \begin{array}{c} i_{\alpha}\\ i_{\beta}\\ \end{array} \right] =\boldsymbol{T}_{3s/2s}\left[ \begin{array}{c} i_a\\ i_b\\ i_c\\ \end{array} \right] [iαiβ]=T3s/2s⎣⎡iaibic⎦⎤
i α = 2 3 [ I 1 cos ( ω e t ) − 1 2 I 1 cos ( ω e t − 2 3 π ) − 1 2 I 1 cos ( ω e t − 4 3 π ) ] = 2 3 [ I 1 cos ( ω e t ) − 1 2 × 2 I 1 cos ( ω e t ) cos ( 2 π 3 ) ] = I 1 cos ( ω e t ) \begin{aligned} i_{\alpha}&=\frac{2}{3}\left[ I_1\cos \left( \omega _et \right) -\frac{1}{2}I_1\cos \left( \omega _et-\frac{2}{3}\pi \right) -\frac{1}{2}I_1\cos \left( \omega _et-\frac{4}{3}\pi \right) \right] \\ &=\frac{2}{3}\left[ I_1\cos \left( \omega _et \right) -\frac{1}{2}\times 2I_1\cos \left( \omega _et \right) \cos \left( \frac{2\pi}{3} \right) \right] \\ &=I_1\cos \left( \omega _et \right) \end{aligned} iα=32[I1cos(ωet)−21I1cos(ωet−32π)−21I1cos(ωet−34π)]=32[I1cos(ωet)−21×2I1cos(ωet)cos(32π)]=I1cos(ωet)
i β = 2 3 [ 3 2 I 1 cos ( ω e t − 2 3 π ) − 3 2 I 1 cos ( ω e t − 4 3 π ) ] = 2 3 × 3 2 × 2 I 1 sin ( ω e t ) sin ( 2 π 3 ) = I 1 sin ( ω e t ) \begin{aligned} i_{\beta}&=\frac{2}{3}\left[ \frac{\sqrt{3}}{2}I_1\cos \left( \omega _et-\frac{2}{3}\pi \right) -\frac{\sqrt{3}}{2}I_1\cos \left( \omega _et-\frac{4}{3}\pi \right) \right] \\ &=\frac{2}{3}\times \frac{\sqrt{3}}{2}\times 2I_1\sin \left( \omega _et \right) \sin \left( \frac{2\pi}{3} \right) \\ &=I_1\sin \left( \omega _et \right) \end{aligned} iβ=32[23I1cos(ωet−32π)−23I1cos(ωet−34π)]=32×23×2I1sin(ωet)sin(32π)=I1sin(ωet)
两相静止坐标系至旋转坐标系的变换矩阵为
T 2 s / 2 r = [ cos ( ω e t ) sin ( ω e t ) − sin ( ω e t ) cos ( ω e t ) ] \boldsymbol{T}_{2s/2r}=\left[ \begin{matrix} \cos \left( \omega _et \right)& \sin \left( \omega _et \right)\\ -\sin \left( \omega _et \right)& \cos \left( \omega _et \right)\\ \end{matrix} \right] T2s/2r=[cos(ωet)−sin(ωet)sin(ωet)cos(ωet)]
即
[ i d i q ] = T 2 s / 2 r [ i α i β ] \left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] =\boldsymbol{T}_{2s/2r}\left[ \begin{array}{c} i_{\alpha}\\ i_{\beta}\\ \end{array} \right] [idiq]=T2s/2r[iαiβ]
i d = cos ( ω e t ) i α + sin ( ω e t ) i β = I 1 \begin{aligned} i_d&=\cos \left( \omega _et \right) i_{\alpha}+\sin \left( \omega _et \right) i_{\beta} \\ &=I_1 \end{aligned} id=cos(ωet)iα+sin(ωet)iβ=I1
i q = − sin ( ω e t ) i α + cos ( ω e t ) i β = 0 \begin{aligned} i_q&=-\sin \left( \omega _et \right) i_{\alpha}+\cos \left( \omega _et \right) i_{\beta} \\ &=0 \end{aligned} iq=−sin(ωet)iα+cos(ωet)iβ=0
若三相电流采用正弦（sin）形式，则各轴系的电流分量为
[ i a i b i c ] = [ I 1 sin ( ω e t ) I 1 sin ( ω e t − 2 π 3 ) I 1 sin ( ω e t − 4 π 3 ) ] \left[ \begin{array}{c} i_a\\ i_b\\ i_c\\ \end{array} \right] =\left[ \begin{array}{c} I_1\sin \left( \omega _et \right)\\ I_1\sin \left( \omega _et-\frac{2\pi}{3} \right)\\ I_1\sin \left( \omega _et-\frac{4\pi}{3} \right)\\ \end{array} \right] ⎣⎡iaibic⎦⎤=⎣⎡I1sin(ωet)I1sin(ωet−32π)I1sin(ωet−34π)⎦⎤
[ i α i β ] = [ I 1 sin ( ω e t ) − I 1 cos ( ω e t ) ] \left[ \begin{array}{c} i_{\alpha}\\ i_{\beta}\\ \end{array} \right] =\left[ \begin{array}{c} I_1\sin \left( \omega _et \right)\\ -I_1\cos \left( \omega _et \right)\\ \end{array} \right] [iαiβ]=[I1sin(ωet)−I1cos(ωet)]
[ i d i q ] = [ 0 − I 1 ] \left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] =\left[ \begin{array}{c} 0\\ -I_1\\ \end{array} \right] [idiq]=[0−I1]
三相坐标系动图代码
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
fig1, ax = plt.subplots()
x = np.arange(0, 2*np.pi, 0.01)
line1, = ax.plot(x, np.cos(x),label="$f_a$")
line2, = ax.plot(x, np.cos(x-2*np.pi/3),label="$f_b$")
line3, = ax.plot(x, np.cos(x-4*np.pi/3),label="$f_c$")
def animate(i):
line1.set_ydata(np.cos(x + i / 50)) # update the data.
line2.set_ydata(np.cos(x + i / 50-2*np.pi/3))
line3.set_ydata(np.cos(x + i / 50-4*np.pi/3))
return line1,line2,line3
ani = animation.FuncAnimation(
fig1, animate, interval=20, blit=True, save_count=50)
plt.legend(loc='upper center', bbox_to_anchor=(0.9, 0.98),shadow=False)
plt.xticks([])
plt.yticks([])
# plt.axis('off')
plt.show()
# export gif
writer = animation.PillowWriter(
fps=15, metadata=dict(artist='Me'), bitrate=1800)
ani.save("three_phase_voltage.gif", writer=writer)
代码运行结果
两相静止坐标系动图源代码
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
fig1, ax = plt.subplots()
x = np.arange(0, 2*np.pi, 0.01)
y1 = 2/3*(np.cos(x)-1/2*np.cos(x-2*np.pi/3)-1/2*np.cos(x-4*np.pi/3))
y2 = 2/3*(np.sqrt(3)/2*np.cos(x-2*np.pi/3)-np.sqrt(3)/2*np.cos(x-4*np.pi/3))
line1, = ax.plot(x, y1 ,label="$f_{\\alpha}$")
line2, = ax.plot(x, y2 ,label="$f_{\\beta}$")
def animate(i):
line1.set_ydata(2/3*(np.cos(x + i / 50)-1/2*np.cos(x + i / 50-2*np.pi/3)-1/2*np.cos(x + i / 50-4*np.pi/3))) # update the data.
line2.set_ydata(2/3*(np.sqrt(3)/2*np.cos(x + i / 50-2*np.pi/3)-np.sqrt(3)/2*np.cos(x + i / 50-4*np.pi/3)))
return line1,line2
ani = animation.FuncAnimation(
fig1, animate, interval=20, blit=True, save_count=50)
plt.legend(loc='upper center', bbox_to_anchor=(0.9, 0.98),shadow=False)
plt.xticks([])
plt.yticks([])
# plt.axis('off')
plt.show()
# export gif
writer = animation.PillowWriter(
fps=15, metadata=dict(artist='Me'), bitrate=1800)
ani.save("clarke_3st2s.gif", writer=writer)
代码运行结果
运动轴系动图源代码
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
fig1, ax = plt.subplots()
x = np.arange(0, 2*np.pi, 0.001)
y5 = np.cos(x)*(2/3*(np.cos(x)-1/2*np.cos(x-2*np.pi/3)-1/2*np.cos(x-4*np.pi/3)))+np.sin(x)*(2/3*(np.sqrt(3)/2*np.cos(x-2*np.pi/3)-np.sqrt(3)/2*np.cos(x-4*np.pi/3)))
y6 = -np.sin(x)*(2/3*(np.cos(x)-1/2*np.cos(x-2*np.pi/3)-1/2*np.cos(x-4*np.pi/3)))+np.cos(x)*(2/3*(np.sqrt(3)/2*np.cos(x-2*np.pi/3)-np.sqrt(3)/2*np.cos(x-4*np.pi/3)))
line1, = ax.plot(x, y5 ,label="$f_d$")
line2, = ax.plot(x, y6 ,label="$f_q$")
def animate(i):
line1.set_ydata(np.cos(x + i / 50)*(2/3*(np.cos(x + i / 50)-1/2*np.cos(x + i / 50-2*np.pi/3)-1/2*np.cos(x + i / 50-4*np.pi/3)))+np.sin(x + i / 50)*(2/3*(np.sqrt(3)/2*np.cos(x + i / 50-2*np.pi/3)-np.sqrt(3)/2*np.cos(x + i / 50-4*np.pi/3)))) # update the data.
line2.set_ydata(-np.sin(x + i / 50)*(2/3*(np.cos(x + i / 50)-1/2*np.cos(x + i / 50-2*np.pi/3)-1/2*np.cos(x + i / 50-4*np.pi/3)))+np.cos(x + i / 50)*(2/3*(np.sqrt(3)/2*np.cos(x + i / 50-2*np.pi/3)-np.sqrt(3)/2*np.cos(x + i / 50-4*np.pi/3))))
return line1,line2
ani = animation.FuncAnimation(
fig1, animate, interval=20, blit=True, save_count=50)
plt.legend(loc='upper center', bbox_to_anchor=(0.9, 0.96),shadow=False)
plt.xticks([])
plt.yticks([])
# plt.axis('off')
plt.show()
# ecxport GIF
writer = animation.PillowWriter(
fps=15, metadata=dict(artist='Me'), bitrate=1800)
ani.save("park_2st2r.gif", writer=writer)
代码运行结果
代码拓展
对于二维物理描述，若能给出时域数学表达式，可以在本文代码的基础上进行修改，替换相应的数学表达式即可获得动图。
参考文献
https://matplotlib.org/stable/gallery/animation/simple_anim.html#sphx-glr-gallery-animation-simple-anim-py