Question 11.10: A 25-kW, 4000-rpm, 220-V, two-pole, three-phase permanent-ma......

a. The rated speed of this machine is

\left(\omega_{\mathrm{m}}\right)_{\mathrm{rated}}=4000\left(\frac{\pi}{30}\right)=419  \mathrm{rad} / \mathrm{sec}

and the rated torque is

T_{\text {rated }}=\frac{P_{\text {rated }}}{\left(\omega_{\mathrm{m}}\right)_{\text {rated }}}=\frac{25 \times 10^{3}}{419}=59.7 \mathrm{~N} \cdot \mathrm{m}

This motor achieves its rated-open-circuit voltage of 220 / \sqrt{3}=127 \mathrm{~V} rms at a speed of n = 3200 r/min. The corresponding electrical frequency is

\omega_{\mathrm{e}}=\left(\frac{\text { poles }}{2}\right)\left(\frac{\pi}{30}\right) n=\left(\frac{\pi}{30}\right) 3200=335  \mathrm{rad} / \mathrm{sec}

From Eq. 11.47,

\Lambda_{\mathrm{PM}}=\frac{\sqrt{2}\left(E_{\text {af }}\right)_{\text {rated }}}{(\omega_{\mathrm{e}})_{\text{rated}}} \quad \quad \quad (11.47)

\Lambda_{\mathrm{PM}}=\frac{\sqrt{2}\left(E_{\text {af }}\right)_{\text {rated }}}{\omega_{\mathrm{e}}}=\frac{\sqrt{2}  127}{335}=0.536 \mathrm{~Wb}

Thus, setting T_{\text {ref }}=0.65 T_{\text {rated }}=38.8 \mathrm{~N} \cdot \mathrm{m}, from Eq. 11.50 we find that

\left(i_{\mathrm{Q}}\right)_{\mathrm{ref}}=\frac{2}{3}\left(\frac{2}{\text { poles }}\right) \frac{T_{\mathrm{ref}}}{\Lambda_{\mathrm{PM}}}=\frac{2}{3}\left(\frac{38.8}{0.536}\right)=48.3 \mathrm{~A}

b. With \left(i_{\mathrm{D}}\right)_{\mathrm{ref}}=0,

\lambda_{\mathrm{D}}=\Lambda_{\mathrm{PM}}=0.536 \mathrm{~Wb}

and

\lambda_{\mathrm{Q}}=L_{\mathrm{s}} i_{\mathrm{Q}}=\left(1.75 \times 10^{-3}\right) 48.3=0.0845 \mathrm{~Wb}

Thus, the rms line-to-neutral armature flux is equal to

\lambda_{\mathrm{a}}=\sqrt{\frac{\lambda_{\mathrm{D}}^{2}+\lambda_{\mathrm{Q}}^{2}}{2}}=\sqrt{\frac{0.536^{2}+0.0845^{2}}{2}}=0.384 \mathrm{~Wb}

The base rms line-to-neutral armature flux can be determined from the base line-to-neutral voltage \left(V_{\mathrm{a}}\right)_{\text {base }}=127 \mathrm{~V} and the base frequency \left(\omega_{\mathrm{e}}\right)_{\text {base }}=419  \mathrm{rad} / \mathrm{sec} as

\left(\lambda_{\mathrm{a}}\right)_{\mathrm{base}}=\frac{\left(V_{\mathrm{a}}\right)_{\text {base }}}{\left(\omega_{\mathrm{e}}\right)_{\text {base }}}=0.303 \mathrm{~Wb}

Thus, the per-unit armature flux is equal to 0.384/0.303 = 1.27 per unit. From this calculation we see that the motor is significantly saturated at this operating condition. In fact, the calculation is probably not accurate because such a degree of saturation will most likely give rise to a reduction in the synchronous inductance as well as the magnetic coupling between the rotor and the stator.

c. In order to maintain rated armature flux linkage, the control will have to produce a direct-axis component of armature current to reduce the direct-axis flux linkage such that the total armature flux linkage is equal to the rated value \left(\lambda_{\mathrm{a}}\right)_{\text {base }}. Specifically, we must have

\lambda_{\mathrm{D}}=\sqrt{2\left(\lambda_{\mathrm{a}}\right)_{\text {base }}^{2}-\lambda_{\mathrm{Q}}^{2}}=\sqrt{2 \times 0.303^{2}-0.0844^{2}}=0.420 \mathrm{~Wb}

We can now find \left(i_{\mathrm{D}}\right)_{\text {ref }} from Eq. 11.48 (setting L_d = L_s)

λ_D=L_di_D+Λ_{PM}\quad \quad \quad (11.48)

\left(i_{\mathrm{D}}\right)_{\mathrm{ref}}=\frac{\lambda_{\mathrm{D}}-\Lambda_{\mathrm{PM}}}{L_{\mathrm{s}}}=\frac{0.420  –  0.536}{1.75 \times 10^{-3}}=-66.3 \mathrm{~A}

The corresponding rms armature current is

I_{\mathrm{a}}=\sqrt{\frac{\left(i_{\mathrm{D}}\right)_{\mathrm{ref}}^{2}+\left(i_{\mathrm{Q}}\right)_{\mathrm{ref}}^{2}}{2}}=\sqrt{\frac{66.3^{2}+48.3^{2}}{2}}=58.0 \mathrm{~A}

The base rms armature current for this machine is equal to

I_{\text {base }}=\frac{P_{\text {base }}}{\sqrt{3}  V_{\text {base }}}=\frac{25 \times 10^{3}}{\sqrt{3}  220}=65.6 \mathrm{~A}

and hence the per-unit armature current is equal to 58.0/65.6 = 0.88 per unit.

Comparing the results of parts (b) and (c) we see how flux weakening by the introduction of direct-axis current can be used to control the terminal voltage of a permanent-magnet synchronous motor under field-oriented control.