Question 10.2: Calculate the per-unit electromagnetic torque versus load an...
Calculate the per-unit electromagnetic torque versus load angle for a saliency ratio between 5 and 50. Lq=0.27 at rated voltage and frequency.
Equation (10.15) can be rearranged in terms of per-unit electromagnetic torque. See also Equation(10.1). The result can be the basis for diagramming Te=f(δs) with saliency ratio as a parameter.
P_{\text {out }}=\omega_{ s } T_{ e }=u_{ s }^{2} \frac{L_{ d }-L_{ q }}{2 L_{ d } L_{ q } \omega_{ s }} \sin 2 \delta_{ s } (10.15)
P=3U^{2}_{sph}\frac{L_{d}-L_{q}}{2\omega _{s}L_{d}L_{q}}\sin 2\delta _{s}=3U^{2}_{sph}\frac{1-\frac{L_{q}}{L_{d}} }{2\omega _{s}L_{q}}=\sin 2\delta _{s} (10.1)
T_{e} = \frac {p_{out}}{w_{s}} = \frac {u^{2}_{s} L_{d} – L_{q}}{w_{s} 2 L_{d}L_{q}w_{s}} sin 2δ_{s} = \frac { u^{2}_{s} 1- \frac{l_{q}}{L_{d}}}{w^{2}_{s} 2 L_{q} } sin 2δ_{s} = 1 \frac { 1- \frac {1}{L_{d}/l_{q}}}{2L_{q} } sin 2δ_{s}For a saliency ratio Ld/Lq=10 and a load angle of 20°
T_{e} = 1 \frac { 1- \frac {1}{L_{d}/l_{q}}}{2L_{q} } sin 2δ_{s} = \frac { 1- \frac{1}{10}}{2⋅0.27} sin 2⋅20° =1.666 sin 40° = 1.07T_{emax10} = 1 \frac { 1- \frac {1}{L_{d}/l_{q}}}{2L_{q} } sin 2δ_{s} = \frac { 1- \frac{1}{10}}{2⋅0.27} sin 2⋅45° =1.666
Therefore, the maximum value for κ=45° is 1.67. Using the same approach, the calculated per-unit torques versus load angles for saliency ratios of Ld/Lq=5,10, and 50 are illustrated in Figure10.5. The maximum values for a saliency ratio of 5 and 50 are as follows.
T_{emax5} = 1 \frac { 1- \frac {1}{L_{d}/l_{q}}}{2L_{q} } sin 2δ_{s} = \frac { 1- \frac{1}{5}}{2⋅0.27} sin 2⋅45° =1.48T_{emax50} = 1 \frac { 1- \frac {1}{L_{d}/l_{q}}}{2L_{q} } sin 2δ_{s} = \frac { 1- \frac{1}{50}}{2⋅0.27} sin 2⋅45° =1.81
Figure10.5 Torque production capacity of a SynRM at different saliency ratios. The q-axis per-unit inductance is constant at Lq=0.27.
Maximum torque is produced when the load angle is δs=π/4. Figure10.5 illustrates how the torque production capacity of a SynRM changes with load angle for the saliency ratios of 5,10, and 50. The q-axis inductance is held constantat Lq=0.27pu.
