\begin{array}{lcl}

\tau=K_t I\\

\omega=K_v V\\

K_t = 1/K_v

\end{array}

$$ To a very rough approximation, these equations are true, and for hobby motors they work quite well. RC vendors usually quote \(K_v\) as the RPM per DC link voltage under trapezoidal commutation.

The above equations model the motor as a speed-dependent voltage source. However, a motor has both inductance and resistance as well. Taking a moment to blatantly ignore the definitions of 'inductance' and 'resistance' (there are several, depending on your conventions), a more accurate voltage equation might be: $$V = \omega/K_v+IR+n_p \omega L$$ Note the intentional lack of subscripts on \(R\) and \(L\); this equation is meant to be heuristic and should not be used to actually compute back EMF.

**The motor equations**

**Taking into account inductance, resistance, and saliency, the complete equations describing a sinusoidally-varying motor with sinusoidal commutation are: $$\begin{array}{lcl}**

\tau=\frac{3}{2}n_p(\lambda I_q+(L_d-L_q)I_d I_q)\\

V_d=R_s I_d-\omega L_q I_q\\

V_q=R_s I_q+\omega L_d I_d+\omega\lambda\\

V_s=\sqrt{V_d^2+V_q^2}

\end{array}

$$ Where \(R_s, L_d\) and \(L_q\) are the resistance and inductance of

**one phase**, \(V_s\) is the peak AC stator voltage across

**one phase**(which for standard SVM is equal to half the DC link voltage), \(\lambda\) is the PM flux linkage, and \(n_p\) is the number of pole pairs. \(I_d\) and \(I_q\) are the usual FOC axis currents.

These equations immediately tell us that IPM's (which have \(L_d < L_q\)) require current on both the d and q-axes to generate the highest torque. This is in stark contrast to SPM's, which typically want \(I_d=0\). On some IPM's, the reluctance component (\((L_d-L_q)I_d I_q\)) is very significant; e.g. for the Hyundai HSG we have \(\lambda=0.053\), \(L_d=0.6 mH\), and \(L_q=1.47 mH\). For high currents, reluctance torque is a huge fraction of the resulting torque...

...as evidenced by this stall test plot, where phase is practically equal to \(3\pi/4\) (the point of highest reluctance torque for a given stator current) at very high currents. This is because reluctance torque grows as the square of current, but PM torque grows only linearly.*

Next time, we will compute the optimum split between the d and q-axis currents for a motor at low speed (one which is not voltage-limited).

*only approximately true because of saturation, but empirical evidence shows it almost works!

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