文章详细介绍了永磁同步电机在高频情况下的电压方程简化过程，利用MATLAB进行方程组求解，并展示了复杂的数学表达式，讨论了如何处理高频响应和电阻压降。通过求解得到d轴和q轴的电流响应，强调了高频响应和轴间耦合的重要性。
一、理论分析
已知永磁同步电机的电机d、q轴电压方程为
在d、q轴分别注入不同频率，不同幅值的高频正弦电压信号
由于在高频下， 此项在电机运行的稳态过程中可以看做是直流项，因此此项可以忽略，则电压方程的高频模型变为
同时，注入电压信号幅值一般为5V~15V，得到的电流响应幅值为0.01pu（额定电流为21A的电机，的高频响应电流幅值仅为0.2A），并且该信号是由多个频率的正弦信号合成的复杂信号，因此电阻压降可以忽略掉，上式简化为
因此该问题成为求解此方程组！
二、MATLAB求解方程组
clear
syms Udh Uqh w1 w2 t
syms Ld Lq we idh iqh
[x,y] = dsolve('Udh*sin(w1*t) = Ld*Didh+we*Lq*iqh', ...
'Uqh*sin(w2*t) = Lq*Diqh-we*Ld*idh','t');
%转为LaTeX格式便于Markdown文件展示
latex(x)
latex(y)
得到的公式为：
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-\frac{\mathrm{Lq}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\left(C_{1}+\frac{\left(-\frac{\mathrm{Udh}\,w_{1}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{1}\right)}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,w_{2}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{2}\right)\,1{}\mathrm{i}}{{w_{2}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Udh}\,\mathrm{we}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{1}\right)\,1{}\mathrm{i}}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,\mathrm{we}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{2}\right)}{{w_{2}}^2-{\mathrm{we}}^2}\right)\,1{}\mathrm{i}}{2\,\mathrm{Lq}}\right)\,1{}\mathrm{i}}{\mathrm{Ld}}+\frac{\mathrm{Lq}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\left(C_{2}+\frac{\left(\frac{\mathrm{Udh}\,w_{1}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{1}\right)}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,w_{2}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{2}\right)\,1{}\mathrm{i}}{{w_{2}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Udh}\,\mathrm{we}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{1}\right)\,1{}\mathrm{i}}{{w_{1}}^2-{\mathrm{we}}^2}-\frac{\mathrm{Uqh}\,\mathrm{we}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{2}\right)}{{w_{2}}^2-{\mathrm{we}}^2}\right)\,1{}\mathrm{i}}{2\,\mathrm{Lq}}\right)\,1{}\mathrm{i}}{\mathrm{Ld}}
−LdLqe−twe1i
C1+2Lq(−w12−we2Udhw1etwe1icos(tw1)+w22−we2Uqhw2etwe1icos(tw2)1i+w12−we2Udhweetwe1isin(tw1)1i+w22−we2Uqhweetwe1isin(tw2))1i
1i+LdLqetwe1i
C2+2Lq(w12−we2Udhw1e−twe1icos(tw1)+w22−we2Uqhw2e−twe1icos(tw2)1i+w12−we2Udhwee−twe1isin(tw1)1i−w22−we2Uqhwee−twe1isin(tw2))1i
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{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\left(C_{1}+\frac{\left(-\frac{\mathrm{Udh}\,w_{1}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{1}\right)}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,w_{2}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{2}\right)\,1{}\mathrm{i}}{{w_{2}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Udh}\,\mathrm{we}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{1}\right)\,1{}\mathrm{i}}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,\mathrm{we}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{2}\right)}{{w_{2}}^2-{\mathrm{we}}^2}\right)\,1{}\mathrm{i}}{2\,\mathrm{Lq}}\right)+{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\left(C_{2}+\frac{\left(\frac{\mathrm{Udh}\,w_{1}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{1}\right)}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,w_{2}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{2}\right)\,1{}\mathrm{i}}{{w_{2}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Udh}\,\mathrm{we}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{1}\right)\,1{}\mathrm{i}}{{w_{1}}^2-{\mathrm{we}}^2}-\frac{\mathrm{Uqh}\,\mathrm{we}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{2}\right)}{{w_{2}}^2-{\mathrm{we}}^2}\right)\,1{}\mathrm{i}}{2\,\mathrm{Lq}}\right)
e−twe1i
C1+2Lq(−w12−we2Udhw1etwe1icos(tw1)+w22−we2Uqhw2etwe1icos(tw2)1i+w12−we2Udhweetwe1isin(tw1)1i+w22−we2Uqhweetwe1isin(tw2))1i
+etwe1i
C2+2Lq(w12−we2Udhw1e−twe1icos(tw1)+w22−we2Uqhw2e−twe1icos(tw2)1i+w12−we2Udhwee−twe1isin(tw1)1i−w22−we2Uqhwee−twe1isin(tw2))1i
可以看出非常复杂
需要采用matlab内置函数进行简化
simplify(x)
simplify(y)
latex(x)
latex(y)
建议参考链接: link
化简之后的公式为
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-\frac{\mathrm{Lq}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\left(C_{1}+\frac{\left(-\frac{\mathrm{Udh}\,w_{1}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{1}\right)}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,w_{2}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{2}\right)\,1{}\mathrm{i}}{{w_{2}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Udh}\,\mathrm{we}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{1}\right)\,1{}\mathrm{i}}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,\mathrm{we}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{2}\right)}{{w_{2}}^2-{\mathrm{we}}^2}\right)\,1{}\mathrm{i}}{2\,\mathrm{Lq}}\right)\,1{}\mathrm{i}}{\mathrm{Ld}}+\frac{\mathrm{Lq}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\left(C_{2}+\frac{\left(\frac{\mathrm{Udh}\,w_{1}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{1}\right)}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,w_{2}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{2}\right)\,1{}\mathrm{i}}{{w_{2}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Udh}\,\mathrm{we}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{1}\right)\,1{}\mathrm{i}}{{w_{1}}^2-{\mathrm{we}}^2}-\frac{\mathrm{Uqh}\,\mathrm{we}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{2}\right)}{{w_{2}}^2-{\mathrm{we}}^2}\right)\,1{}\mathrm{i}}{2\,\mathrm{Lq}}\right)\,1{}\mathrm{i}}{\mathrm{Ld}}
−LdLqe−twe1i
C1+2Lq(−w12−we2Udhw1etwe1icos(tw1)+w22−we2Uqhw2etwe1icos(tw2)1i+w12−we2Udhweetwe1isin(tw1)1i+w22−we2Uqhweetwe1isin(tw2))1i
1i+LdLqetwe1i
C2+2Lq(w12−we2Udhw1e−twe1icos(tw1)+w22−we2Uqhw2e−twe1icos(tw2)1i+w12−we2Udhwee−twe1isin(tw1)1i−w22−we2Uqhwee−twe1isin(tw2))1i
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{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\left(C_{1}+\frac{\left(-\frac{\mathrm{Udh}\,w_{1}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{1}\right)}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,w_{2}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{2}\right)\,1{}\mathrm{i}}{{w_{2}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Udh}\,\mathrm{we}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{1}\right)\,1{}\mathrm{i}}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,\mathrm{we}\,{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{2}\right)}{{w_{2}}^2-{\mathrm{we}}^2}\right)\,1{}\mathrm{i}}{2\,\mathrm{Lq}}\right)+{\mathrm{e}}^{t\,\mathrm{we}\,1{}\mathrm{i}}\,\left(C_{2}+\frac{\left(\frac{\mathrm{Udh}\,w_{1}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{1}\right)}{{w_{1}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Uqh}\,w_{2}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\cos\left(t\,w_{2}\right)\,1{}\mathrm{i}}{{w_{2}}^2-{\mathrm{we}}^2}+\frac{\mathrm{Udh}\,\mathrm{we}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{1}\right)\,1{}\mathrm{i}}{{w_{1}}^2-{\mathrm{we}}^2}-\frac{\mathrm{Uqh}\,\mathrm{we}\,{\mathrm{e}}^{-t\,\mathrm{we}\,1{}\mathrm{i}}\,\sin\left(t\,w_{2}\right)}{{w_{2}}^2-{\mathrm{we}}^2}\right)\,1{}\mathrm{i}}{2\,\mathrm{Lq}}\right)
e−twe1i
C1+2Lq(−w12−we2Udhw1etwe1icos(tw1)+w22−we2Uqhw2etwe1icos(tw2)1i+w12−we2Udhweetwe1isin(tw1)1i+w22−we2Uqhweetwe1isin(tw2))1i
+etwe1i
C2+2Lq(w12−we2Udhw1e−twe1icos(tw1)+w22−we2Uqhw2e−twe1icos(tw2)1i+w12−we2Udhwee−twe1isin(tw1)1i−w22−we2Uqhwee−twe1isin(tw2))1i
由于matlab并不能很好的按照想要的方式合并公式中的同类项，还需要手动化简：
对这个方程组进行求解，可以得到d轴方程的通解为
d轴方程的特解为
解得q轴方程的通解为
Q轴方程的特解为
总结
dq轴的两个初值项在同步旋转轴系下是直流项，旋转方向不同，特解项包含高频响应分量以及dq轴之间的耦合响应分量。因此，与初值有关的通解可以忽略，可以得到
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