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${V}_{a}=\sqrt{\frac{{v}_{D}^{2}+{v}_{Q}^{2}}{2}}=\sqrt{\frac{({R}_{a}{i}_{D}-{\omega}_{e}{\lambda}_{Q}{)}^{2}+({R}_{a}{i}_{Q}+{\omega}_{e}{\lambda}_{D}{)}^{2}}{2}}$
$=\sqrt{\frac{{\left[{R}_{a}{i}_{D}-{\omega}_{e}\left[{L}_{S}-\frac{{L}_{m}^{2}}{{L}_{R}}\right]{i}_{Q}\right]}^{2}+({R}_{a}{i}_{Q}+{\omega}_{e}{L}_{S}{i}_{D}{)}^{2}}{2}}$ (10.87)
These equations show that the armature flux linkages and terminal voltage are determined by both the direct- and quadrature-axis components of the armature current.
Thus, the block marked "Auxiliary Controller" in Fig. 10.20 a, which calculates the reference values for the direct- and quadrature-axis currents, must calculate the reference currents $({i}_{D}{)}_{\text{ref}}$ and $({i}_{Q}{)}_{\text{ref}}$ which achieve the desired torque subject to constraints on armature flux linkages (to avoid saturation in the motor), armature current, $({I}_{a}{)}_{\text{rms}}=\sqrt{({i}_{D}^{2}+{i}_{Q}^{2})/2}$ (to avoid excessive armature heating) and armature voltage (to avoid potential insulation damage).
Note that, as we discussed with regard to synchronous machines in Section 10.2.2, the torque-control system of Fig. 10.20 a is typically imbedded within a larger control loop. One such example is the speed-control loop of Fig. 10.20 b.
The ability to independently control the rotor flux and the torque has important control implications. Consider, for example, the response of the direct-axis rotor flux to a change in direct-axis current. Equation C.66, with ${\lambda}_{\text{qR}}=0$ , becomes
$0={R}_{\text{aR}}{i}_{\text{dR}}+\frac{{\mathrm{d\lambda}}_{\text{dR}}}{\text{dt}}$ (10.88)
Substituting for ${i}_{\text{dR}}$ in terms of ${\lambda}_{\text{dR}}$
${i}_{\text{dR}}=\frac{{\lambda}_{\text{dR}}-{L}_{m}{i}_{d}}{{L}_{R}}$ (10.89)
gives a differential equation for the rotor flux linkages ${\lambda}_{\text{DR}}$
$\frac{{\mathrm{d\lambda}}_{\text{dR}}}{\text{dt}}+\left[\frac{{R}_{\text{aR}}}{{L}_{R}}\right]{\lambda}_{\text{dR}}=\left[\frac{{L}_{m}}{{L}_{R}}\right]{i}_{d}$ (10.90)
From Eq. 10.90 we see that the response of the rotor flux to a step change in direct axis current id is relatively slow; ${\lambda}_{\text{dR}}$ will change exponentially with the rotor time constant of ${\tau}_{R}={L}_{R}/{R}_{\text{aR}}$ . Since the torque is proportional to the product ${\lambda}_{\text{dR}}{i}_{q}$ we see that fast torque response will be obtained from changes in ${i}_{q}$ . Thus, for example, to implement a step change in torque, a practical control algorithm might start with a step change in $({i}_{Q}{)}_{\text{ref}}$ to achieve the desired torque change, followed by an adjustment in $({i}_{D}{)}_{\text{ref}}$ (and hence ${\lambda}_{\text{dR}}$ ) to readjust the armature current or terminal voltage as desired. This adjustment in $({i}_{D}{)}_{\text{ref}}$ would be coupled with a compensating adjustment in $({i}_{Q}{)}_{\text{ref}}$ to maintain the torque at its desired value.
Unlike dc and ac (synchronous or induction) machines, VRMs cannot be simply "plugged in" to a dc or ac source and then be expected to run. As is dicussed in Chapter 8, the phases must be excited with (typically unipolar) currents, and the timing of these currents must be carefully correlated with the position of the rotor poles in order to produce a useful, time-averaged torque. The result is that although the VRM itself is perhaps the simplest of rotating machines, a practical VRM drive system is relatively complex.
VRM drive systems are competitive only because this complexity can be realized easily and inexpensively through power and microelectronic circuitry. These drive systems require a fairly sophisticated level of controllability for even the simplest modes of VRM operation. Once the capability to implement this control is available, fairly sophisticated control features can be added (typically in the form of additional software) at little additional cost, further increasing the competitive position of VRM drives.
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