Question 10.5: Derive(10.37). idref=ψsmaxref/√2La
Derive (10.37).
i_{dref} = \frac { Ψ_{smaxref}}{ \sqrt{2} L_{d} }√
Differentiating the torque equation
T_{e} = \frac {3}{2} p ( \frac { L_{d} }{L_{q} } – 1 ) i_{d} \sqrt { Ψ^{2}_{s} – ( L_{d} i_{d} ) ^{2} }with respect to id, results in
\frac { ∂T_{e} }{ ∂i_{d} } = \frac {3}{2} p( \frac { L_{d} }{L_{q} } – 1 ) [\sqrt { Ψ^{2}_{s} – ( L_{d} i_{d} ) ^{2} } + i_{d} \frac { -2L^{2}_{d} i_{d} } { 2 \sqrt { Ψ^{2}_{s} – ( L_{d} i_{d} ) ^{2} } } ]The zero for the differential can be found by setting the bracketed term to zero.
[\sqrt { Ψ^{2}_{s} – ( L_{d} i_{d} ) ^{2} } + i_{d} \frac { -2L^{2}_{d} i_{d} } { 2 \sqrt { Ψ^{2}_{s} – ( L_{d} i_{d} ) ^{2} } } ] = 0This results in
[ \frac { Ψ^{2}_{s} – 2( L_{d} i_{d} ) ^{2} }{ \sqrt { Ψ^{2}_{s} – ( L_{d} i_{d} ) ^{2} } } ] =0The numerator becomes 0 when
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