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Field Oriented Control

System model

The governing equation of a salient-pole permanent magnet synchronous motor is expressed as
vdq=Ridq+ddtϕi+ωreJϕi+ωreJϕmωm=P−1ωreτm=P(Φiq+(Ld−Lq)idiq),\begin{align} \bm{v}_{\rm dq}&=\bm{R}\bm{i}_{\rm dq}+\frac{d}{dt}\bm{\phi}_{\rm i}+\omega_{\rm re}\bm{J}\bm{\phi}_{\rm i}+\omega_{\rm re}\bm{J}\bm{\phi}_{\rm m}\\ \omega_{\rm m}&=P^{-1}\omega_{\rm re}\\ \tau_{\rm m}&=P\left(\Phi i_{\rm q}+ (L_{\rm d}-L_{\rm q })i_{\rm d}i_{\rm q}\right), \end{align}
where R\bm{R}, Ldq\bm{L}_{\rm dq}, vdq\bm{v}_{\rm dq}, idq\bm{i}_{\rm dq}, ϕi\bm{\phi}_{\rm i}, ϕm\bm{\phi}_{\rm m}, ωre\omega_{\rm re}, ωm\omega_{\rm m}, τm\tau_{\rm m}, and PP stand for an armature resistance, inductance on the dq-axis, voltage and current on dq-axis, armature reaction flux, rotor flux, electric and mechanical angular speed of a motor, motor torque, and the number of motor poles, expressed as
R=RI, Ldq=diag(Ld, Lq),vdq=[vdvq], idq=[idiq], ϕi=Ldqidq, ϕm=[Φ0].\begin{align} \bm{R}&=R\bm{I},\ \bm{L}_{\rm dq}={\rm diag}(L_{\rm d},\ L_{\rm q}), \notag\\ \bm{v}_{\rm dq}&=\begin{bmatrix} v_{\rm d} \\ v_{\rm q} \end{bmatrix},\ \bm{i}_{\rm dq}=\begin{bmatrix} i_{\rm d} \\ i_{\rm q} \end{bmatrix},\ \bm{\phi}_{\rm i}=\bm{L}_{\rm dq}\bm{i}_{\rm dq},\ \bm{\phi}_{\rm m}=\begin{bmatrix} \Phi \\ 0\end{bmatrix}. \notag \end{align}
A magnetic flux Φ\Phi is generated by a permanent magnet and gets constant value. The matrices I\bm{I} and J\bm{J} denote an identity matrix and rotation operator expressed as
J=[0−110].\begin{align} \bm{J}&=\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}. \end{align}

Field-oriented control

Assuming that the initial values of all the state are zero, we get an equation in a Laplace domain
sLdqidq=−Ridq−ωreJ(ϕi+ϕm)+vdq,\begin{align} s\bm{L}_{\rm dq}\bm{i}_{\rm dq} &= -\bm{R}\bm{i}_{\rm dq}-\omega_{\rm re}\bm{J}(\bm{\phi}_{\rm i}+\bm{\phi}_{\rm m})+\bm{v}_{\rm dq}, \end{align}
where ss stands for a complex value. Here, let us design an input reference vdqref\bm{v}^{\rm ref}_{\rm dq} as
vdqref=Ci(idqcmd−idq)+ω^reJ(ϕ^i+ϕ^m)Ci≡(L^dq+s−1R^)ωcωc≡diag(ωc,d, ωc,q),\begin{align} \bm{v}^{\rm ref}_{\rm dq}&=\bm{C}_{\rm i}(\bm{i}_{\rm dq}^{\rm cmd}-\bm{i}_{\rm dq})+\hat{\omega}_{\rm re}\bm{J}(\hat{\bm{\phi}}_{\rm i}+\hat{\bm{\phi}}_{\rm m})\\ \bm{C}_{\rm i}&\equiv (\hat{\bm{L}}_{\rm dq}+s^{-1}\hat{\bm{R}})\bm{\omega}_{\rm c}\\ \bm{\omega}_{\rm c}&\equiv{\rm diag}(\omega_{\rm c,d},\ \omega_{\rm c,q}), \end{align}
where Ci\bm{C}_{\rm i} and ωc\bm{\omega}_{\rm c} are a PI controller and controller gain that determines the natural frequency of a current controller, and the superscript cmd^{\rm cmd}, ref^{\rm ref} and ^\hat{} stand for a command, reference, and estimated values. If a modeling error is sufficiently small, this feedback system controls the current idq\bm{i}_{\rm dq} as it gets
idq=(sI+ωc)−1ωcidqcmd.\begin{align} \bm{i}_{\rm dq}=(s\bm{I}+\bm{\omega}_{\rm c})^{-1}\bm{\omega}_{\rm c}\bm{i}_{\rm dq}^{\rm cmd}. \end{align}
By setting idqcmd=[0Φ−1P−1τmcmd]T\bm{i}^{\rm cmd}_{\rm dq}=\begin{bmatrix}0&\Phi^{-1}P^{-1}\tau_{\rm m}^{\rm cmd}\end{bmatrix}^{\mathrm T}, a motor torque τm\tau_{\rm m} is controlled as
τm=Gcτmcmd,\begin{align} \tau_{\rm m}&=G_{\rm c}\tau_{\rm m}^{\rm cmd}, \end{align}
where GcG_{\rm c} represents first-order lag characteristics
Gc=(s+ωc,q)−1ωc,q.\begin{align} G_{\rm c}&=(s+\omega_{\rm c,q})^{-1}\omega_{\rm c,q}. \end{align}
Thus, current control on the dq-axis enables torque control.

CONTENTS

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