## 4 Three-Phase Inverters
### Control of PMSM  
This section presents a typical control scheme for Permanent Magnet Synchronous Motors (PMSM). These motors are synchronous machines, in which permanent magnets are embedded in the rotor to create a constant magnetic field. As in all synchronous machines, at steady state the rotor speed is proportional to the frequency of currents and voltages in the stator. For this reason, such motors are especially useful in applications that require precise speed or position control. This section focuses on three-phase PMSM that have sinusoidal EMF (as opposed to motors with trapezoidal EMF). Permanent magnet synchronous motors (with sinusoidal EMF) are modeled like synchronous generators, with three modifications:  
* The stator currents are defined positive when flowing into the machine. * The term \(L_{q}i_{f}\) is replaced with \(\lambda\), which is the amplitude of the flux induced in the stator phases by the permanent magnets on the rotor. * The electromagnetic torque accelerates the rotor, and the mechanical torque decelerates the rotor. The angular acceleration is defined as \(\frac{\mathrm{d}}{\mathrm{d}t}\omega_{m}=\frac{1}{f}\left(T_{e}-T_{m}\right)\). The resulting model is  
\[\frac{\mathrm{d}}{\mathrm{d}t}\theta =p\omega_{m}, \tag{159}\] \[\frac{\mathrm{d}}{\mathrm{d}t}\omega_{m} =\frac{1}{f}\left(T_{e}-T_{m}\right),\] \[\frac{\mathrm{d}}{\mathrm{d}t}i_{d} =\frac{1}{L_{d}}v_{d}-\frac{R}{L_{d}}i_{d}+\frac{L_{q}}{L_{d}}p \omega_{m}i_{d},\] \[\frac{\mathrm{d}}{\mathrm{d}t}i_{q} =\frac{1}{L_{q}}v_{q}-\frac{R}{L_{q}}i_{q}-\frac{L_{d}}{L_{q}}p \omega_{m}i_{d}-\frac{\lambda p\omega_{m}}{L_{q}},\] \[\frac{\mathrm{d}}{\mathrm{d}t}i_{0} =\frac{1}{L_{0}}v_{0}-\frac{R}{L_{0}}i_{0}.\]  
The inputs of the model are \(v_{d}\), \(v_{q}\), \(v_{0}\) and \(T_{m}\), and several additional outputs are  
\[T_{e} =\frac{3}{2}p\left(\lambda i_{q}+\left(L_{d}-L_{q}\right)i_{d}i_{q }\right), \tag{160}\] \[p_{m} =T_{m}\omega_{m},\quad p_{e}=T_{e}\omega_{m}.\]  
The reference angle for the \(dq0\) transformation is the electrical angle \(\theta\). The symbols appearing in these equations are:  
* \(\theta\) is the rotor electrical angle, measured with respect to a fixed point on the stator;
* \(p=\mathrm{poles}/2\) is the number of pole pairs;
* \(L_{d}\), \(L_{q}\), \(L_{0}\) are the direct-axis, quadrature-axis, and zero-sequence inductances;
* \(R\) is the resistance of the stator windings;
* \(i_{d}\), \(i_{q}\), \(i_{0}\) are the stator currents (positive when flowing into the machine);
* \(v_{d}\), \(v_{q}\), \(v_{0}\) are the stator voltages;
* \(\omega_{m}\) is the angular velocity of the rotor;
* \(\lambda\) is the amplitude of the flux induced in the stator phases by the permanent magnets on the rotor;
* \(J\) is the rotor moment of inertia;
* \(T_{m}\), \(T_{e}\) are the mechanical and electromagnetic torques;
* \(p_{m}\), \(p_{e}\) are the mechanical and electromagnetic powers. Similarly to synchronous generators, the model may be simplified by assuming a round rotor (no saliency effects), such that \(L_{d}=L_{q}=L_{s}\). In this case, the motor may be described by the equivalent circuit in Figure 34.  
The induced EMF is given by \(e_{d}=0\), \(e_{q}=\lambda p\omega_{m}\), \(e_{0}=0\), where \([e_{d},e_{q},e_{0}]^{\mathsf{T}}\) is the \(dq0\) transformation of \([e_{a},e_{b},e_{c}]^{\mathsf{T}}\). In addition, for \(L_{d}=L_{q}=L_{s}\), the electromagnetic torque is given by  
\[T_{e}=\frac{3}{2}p\lambda i_{q}. \tag{161}\]  
Therefore, the induced EMF is proportional to the angular velocity, while the electric torque is proportional to the quadrature-axis current. The heart of the energy conversion process is described by the induced EMF source, which converts electrical energy to mechanical energy. From a mechanical perspective, the electromagnetic power is  
\[p_{e}=T_{e}\omega_{m}=\frac{3}{2}p\lambda\omega_{m}i_{q}, \tag{162}\]  
and from an electrical perspective,  
\[\frac{\mathrm{d}}{\mathrm{d}t}\omega_{m} =\frac{1}{f}\left(T_{e}-T_{m}\right),\]0  
Both expressions are identical.