The Motor Equations

We are all taught as wee seedlings that motors, brushed or brushless, are governed by the following equations: $$
\begin{array}{lcl}
\tau=K_t I\\
\omega=K_v V\\
K_t = 1/K_v
\end{array}
$$ 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!