Speed and torque control  (Page 14/19)

In a practical implementation of the technique which we have derived, the directand quadrature-axis currents $i_D\text{and}{i}_Q$ must be transformed into the three motor phase currents $i_a(t),i_b(t),\text{and}{i}_c(t)$ . This can be done using the inverse dq0 transformation of Eq. C.48 which requires knowledge of ${\theta }_S$ , the electrical angle between the axis of phase a, and the direct-axis of the synchronously rotating reference frame.

Since it is not possible to measure the axis of the rotor flux directly, it is necessary to calculate ${\theta }_S$ , where ${\theta }_S={\omega }_{e}t+{\theta }_0$ as given by Eq. C.46. Solving Eq. 10.70 for ${\omega }_e$ gives ${\omega }_e={\omega }_{\text{me}}-R_{\text{aR}}\left[\frac{i_{\text{QR}}}{{\lambda }_{\text{DR}}}\right]$ (10.79)

From Eq. 10.63 with ${\lambda }_{\text{QR}}$ = 0 we see that

$i_{\text{QR}}=-\left[\frac{L_m}{L_R}\right]i_Q$ (10.80)

Eq. 10.80 in combination with Eq. 10.77 then gives

${\omega }_e={\omega }_{\text{me}}+\frac{R_{\text{aR}}}{L_R}\left[\frac{i_Q}{i_D}\right]={\omega }_{\text{me}}+\frac{1}{{\tau }_R}\left[\frac{i_Q}{i_D}\right]$ (10.81)

where ${\tau }_R=L_R/R_{\text{aR}}$ is the rotor time constant. We can now integrate Eq. 10.81 to find

${\hat{\theta }}_S=\left[{\omega }_{\text{me}}+\frac{1}{{\tau }_R}\left[\frac{i_Q}{i_D}\right]\right]t+{\theta }_0$ (11.82)

where ${\hat{\theta }}_S$ indicates the calculated value of ${\theta }_S$ (often referred to as the estimated value of ${\theta }_S$ ). In the more general dynamic sense

${\hat{\theta }}_S={\int }_o^t\left[{\omega }_{\text{me}}+\frac{1}{{\tau }_R}\left[\frac{i_Q}{i_D}\right]\right]d{t}^{\text{'}}+{\theta }_0$ (10.83)

Note that both Eqs. 10.82 and 10.83 require knowledge of ${\theta }_0$ , the value of ${\theta }_S$ at t = 0. Although we will not prove it here, it turns out that in a practical implementation, the effects of an error in this initial angle decay to zero with time, and hence it can be set to zero without any loss of generality.

Figure 10.20 a shows a block diagram of a field-oriented torque-control system for an induction machine. The block labeled "Estimator" represents the calculation of Eq. 10.83 which calculates the estimate of ${\theta }_S$ required by the transformation from dq0 to abc variables.

Note that a speed sensor is required to provide the rotor speed measurement required by the estimator. Also notice that the estimator requires knowledge of the rotor time constant ${\tau }_R=L_R/R_{\text{aR}}$ . In general, this will not be known exactly, both due to uncertainty in the machine parameters as well as due to the fact that the rotor

Figure 10.20 (a) Block diagram of a field-oriented torque-control system

for an induction motor. (b) Block diagram of an induction-motor speed-control

loop built around a field-oriented torque control system.

resistance $R_{\text{aR}}$ will undoubtedly change with temperature as the motor is operated. It can be shown that errors in ${\tau }_R$ result in an offset in the estimate of ${\theta }_S$ , which in turn will result in an error in the estimate for the position of the rotor flux with the result that the applied armature currents will not be exactly aligned with the direct- and quadrature-axes. The torque controller will still work basically as expected, although there will be corresponding errors in the torque and rotor flux.

As with the synchronous motor, the rms armature flux-linkages can be found from Eq. 10.36 as

$({\lambda }_a{)}_{\text{rms}}=\sqrt{\frac{{\lambda }_D^2+{\lambda }_Q^2}{2}}$ (10.84)

Combining Eqs. 10.61 and 10.80 gives

${\lambda }_Q=L_S{i}_Q+L_m{i}_{\text{QR}}=\left[L_S-\frac{L_m^2}{L_R}\right]i_Q$ (10.85)

Substituting Eqs. 10.78 and 10.85 into Eq. 11.84 gives

$({\lambda }_a{)}_{\text{rms}}=\sqrt{\frac{L_S^2{i}_D^2+{\left[L_s-\frac{L_m^2}{L_R}\right]}^2{i}_Q^2}{2}}$ (10.86)

Finally, as discussed in the footnote to Eq. 10.35, the rms line-to-neutral armature voltage can be found as