PCS的dq坐标系控制方程化简推导

αβ坐标系下的控制方程为：
Uαβ – Eαβ = RIαβ + Ld(Iαβ)/dt
Uαβ – Eαβ - RIαβ = Ld(Iαβ)/dt

令 Uαβ – Eαβ - RIαβ = Xαβ
有： Xαβ = Ld(Iαβ)/dt

根据dq逆变换公式 Xαβ = [cosθ -sinθ ] [Xd]
[sinθ cosθ ] [Xq] 得到

Xα = Xdcosθ – Xqsinθ
Xβ = Xdsinθ + Xqcos

带入 Xαβ = Ld(Iαβ)/dt 得到
Xdcosθ – Xqsinθ = Ld(Idcosθ)/dt – L d(Iqsinθ)/dt
Xdsinθ + Xqcos = Ld(Idsinθ)/dt + L d(Iqcosθ)/dt
把θ = wt带入上式得到：
Xdcos(wt) – Xqsin(wt) = Ld(Idcos(wt))/dt – L d(Iqsin(wt))/dt
Xdsin(wt) + Xqcos(wt) = Ld(Idsin(wt))/dt + L d(Iq*cos(wt))/dt

微分化简得到：
Ld(Idcos(wt))/dt = L*[ cos(wt)d(Id)/dt – wsin(wt)Id ]
L d(Iqsin(wt))/dt = L[ sin(wt)d(Iq)/dt + wcos(wt)*Iq ]

Ld(Idsin(wt))/dt = L*[sin(wt)d(Id)/dt + wcos(wt)Id]
Ld(Iqcos(wt))/dt = L[cos(wt)d(Iq)/dt –wsin(wt)*Iq]

带入前面的式子得到：
Xdcos(wt)–Xqsin(wt) = L*[ cos(wt)d(Id)/dt – wsin(wt)Id - sin(wt)d(Iq)/dt - wcos(wt)Iq]
Xdsin(wt)+Xqcos(wt) = L*[ sin(wt)d(Id)/dt + wcos(wt)*Id + cos(wt)d(Iq)/dt –wsin(wt)*Iq]

以上两个式子分别同除以sin(wt)和cos(wt),并相加得到：
(cos(wt)/sin(wt) + sin(wt)/cos(wt))Xd = L(cos(wt)/sin(wt) + sin(wt)/cos(wt))( d(Id)/dt – wIq))
Xd = Ld(Id)/dt – LwIq
Ud – Ed - RId = Xd代入上式得到：

Ud – Ed = RId + Ld(Id)/dt – LwIq
同理做相减化简得到
Uq – Eq = RIq + Ld(Iq)/dt + LwId
由于稳态的时候频率恒定，w = w0，上两个式子化简为：
Ud – Ed = RId + Ld(Id)/dt – Lw0Iq
Uq – Eq = RIq + Ld(Iq)/dt + Lw0Id
可见上个方程，dq相互耦合，具体工程是采用前馈解耦的方式解决