Energy Performance of Brushless DC Motors with a Sinusoidal EMF Waveform

E. R. Geints, Yu. N. Kroneberg

(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)

As is known, one of the main features of brushless DC motors is associated with the nonsinusoidal form of the phase voltages. With a nonsinusoidal rotor field this leads to the appearance of additional rotating and pulsating torques. Studies show [1, equation (18)] that the higher time harmonics of the current, while increasing the power consumed, by no means always create a corresponding increase in useful torque (because of the lower efficiency, and also because of the counteracting effect of some harmonics).

Let us determine the conditions for a favorable combination of the forms of the phase current and EMF for neutral commutation \(\left(\gamma_0 = \dfrac{\pi}{2}\right)\). In accordance with the equations obtained earlier [1, equations (18) and (19)], the efficiencies of the first harmonic, of the harmonics that create forward and reverse rotating fields, and also of pulsating fields are described by the expressions:

\[ \eta_1 = \frac{M_{\mathrm{vr}1}\Omega}{P_{1\mathrm{potr}1}} = \frac{\Omega}{\Omega_0} = \nu, \]

\[ \eta_{(2mn+1)} = \nu \frac{E_{M(2mn+1)}\,U_{M1}} {E_{M1}\,U_{M(2mn+1)}}, \tag{1} \]

\[ \eta_{(2mn-1)} = \nu \frac{E_{M(2mn-1)}\,U_{M1}} {E_{M1}\,U_{M(2mn-1)}}, \]

\[ \eta_{[m(2n-1)]} = \nu \frac{E_{M[m(2n-1)]}\,U_{M1}} {E_{M1}\,U_{M[m(2n-1)]}}, \]

where the notation has been adopted in accordance with [1].

It follows from equation (1) that the maximum efficiency of each harmonic and, consequently, the overall efficiency of the motor are numerically equal to the relative speed \(\nu\) (theoretically it can reach unity). For this, complete identity of the forms of the phase currents and EMF is required. If the latter condition is satisfied, the efficiency and utilization of the machine will always be the same, irrespective of the content of higher harmonics.

The form of the phase current is, naturally, determined by the form of the voltage (the winding connection scheme and the duty factor \(k\) of operation of the power elements of the commutator), and also by the inductive reactances of the winding for each harmonic, whose magnitude depends on the rotational speed. Therefore, in the general case the current waveform does not remain constant, and as the rotational speed increases it is smoothed and the content of higher harmonics is reduced. The only exception is a sinusoidal current waveform.

The form of the phase emf is determined by the form of the excitation field, i.e., by the spectrum of the field harmonics and the corresponding winding factors. As is known, in most cases two-layer distributed windings with a shortened pitch, characterized by practically sinusoidal emf even with substantial distortions of the field, are the most effective with respect to the use of active materials. Concentrated windings with a diametral pitch, whose emf contains all odd field harmonics, are used in ac machines mainly for technological reasons and, as a rule, have poorer utilization and increased reactances. In general, therefore, notwithstanding possible exceptions, a sinusoidal emf is the most rational for electrical machines. From these positions it is of interest to analyze the energy characteristics of brushless dc motors with a sinusoidal emf.

With neutral commutation, using the previously obtained equations of torques and powers [1, equations (18) and (19)], and taking without significant error \(X_{\text{e}}=X_0\), we find the general expression for the efficiency of a machine with sinusoidal emf:

\[ \eta_{\Sigma}= \frac{M_{\text{vr}}\Omega}{P_{1\text{cons.}}} = \frac{ \nu\,\dfrac{1-\nu}{1+\tau_0^2\nu^2} }{ \left\{ \dfrac{1-\nu}{1+\tau_0^2\nu^2} + \sum_{n=1}^{\infty} \frac{U^2_{M(2mn+1)}}{\overline{U}_{M1}^{\,2}\left[1+(2mn+1)^2\tau_0^2\nu^2\right]} + \right. } \;\to \]

\[ \to\; \frac{1}{ \left. + \sum_{n=1}^{\infty} \frac{U^2_{M(2mn-1)}}{\overline{U}_{M1}^{\,2}\left[1+(2mn-1)^2\tau_0^2\nu^2\right]} + \sum_{n=1}^{\infty} \frac{U^2_{M[m(2n-1)]}}{\overline{U}_{M1}^{\,2}\left[1+m^2(2n-1)^2\tau_0^2\nu^2\right]} \right\} }. \tag{2} \]

For specific winding schemes this expression is considerably simplified. Thus, with independent supply of the phases [1, Fig. 1, a],

\[ \eta_{\Sigma}^{(\text{н})}= \]

\[ = \frac{ \nu\left[ \dfrac{1-\nu}{1+\tau_0^2\nu^2} \right] }{ \left\{ \dfrac{\pi^2}{8}\, \frac{k}{\sin^2\frac{\pi}{2}\,k} - \frac{\pi\tau_0\nu}{4\sin^2\frac{\pi}{2}\,k} \left[ \frac{ \operatorname{ch}\dfrac{\pi}{2\tau_0\nu}(1-k)\operatorname{sh}\dfrac{\pi k}{2\tau_0\nu} }{ \operatorname{ch}\dfrac{\pi}{2\tau_0\nu} } \right] - \frac{\nu}{1+\tau_0^2\nu^2} \right\} }. \tag{3} \]

and, when the machine windings are connected in “star” or “delta,”

\[ \eta_2^{(\nabla,\lambda)} = \frac{ \nu \left[\dfrac{1-\nu}{1+\tau_0^2\nu^2}\right] }{ \left\{ \dfrac{\pi^2}{36}\,\dfrac{(3k+1)}{\sin^2 \dfrac{\pi}{2}k} - \dfrac{\pi \tau_0 \nu}{12 \sin^2 \dfrac{\pi}{2}k} \left[ \dfrac{ 3\operatorname{ch}\dfrac{\pi(1-k)}{2\tau_0\nu}\operatorname{sh}\dfrac{\pi k}{2\tau_0\nu} }{ \operatorname{ch}\dfrac{\pi}{2\tau_0\nu} } - \right.\right. \]
\[ \left.\left. - \dfrac{ \operatorname{ch}\dfrac{\pi(1-k)}{2\tau_0\nu}\operatorname{sh}\dfrac{\pi(3k-2)}{6\tau_0\nu} }{ \operatorname{ch}\dfrac{\pi}{6\tau_0\nu} } \right] - \dfrac{\nu}{1+\tau_0^2\nu^2} \right\} }. \tag{4} \]

Figure 1 shows the dependences of the efficiency on rotational speed, calculated from equations (3) and (4), for various values of the relative inductance \(\tau_0=\left(\dfrac{X_a}{R_s}\,\dfrac{\Omega_0}{\Omega}\right)\) for the winding-connection circuits under consideration. The limiting curve obtained under the assumption of sinusoidal voltage and current is also given there, i.e., with complete identity of the current and emf waveforms. As follows from Fig. 1, increasing the relative inductance raises the efficiency, which is explained by the filtering out of higher current harmonics. However, on this basis one still cannot conclude that it is advisable to increase \(\tau_0\), because of the simultaneous reduction in the utilization of the machine, which can be characterized quite fully by the ratio of the useful power for sinusoidal waveforms at a given value of \(\tau_0\) \((q_2(\tau_0\ne0))\) to its limiting value at \(\tau_0=0\). This ratio can be obtained from the torque equation [1, equation (18)] and is described by the simple function

\[ k_{q_2}=\frac{q_2(\tau_0\ne0)}{q_2(\tau_0=0)}= \frac{1}{1+\tau_0^2\nu^2}. \tag{5} \]

Figures 2 and 3 show the dependences of the maximum efficiency and the efficiency at maximum useful power on the relative inductance, i.e., efficiencies that practically bound the operating region, as well as the dependences of the coefficient \(k_{q_2}\) for the relative speeds corresponding to the maximum values of efficiency and maximum useful powers. These curves already give a more complete picture of the role of relative inductance and make it possible, from a known value of \(\tau_0\) (which is ultimately determined by the type and dimensions of the machine), to estimate the change in efficiency. Already at \(\tau_0 \ge 0.5\), the reduction in efficiency due to a certain mismatch between the current and emf waveforms becomes insignificant, and therefore in this case the sinusoidal emf waveform may be regarded as optimal. For smaller values of \(\tau_0\), additional investigations are probably necessary that would make it possible to compare the possible gain due to better approximation of the emf waveform to the current waveform with the loss caused by obtaining a nonsinusoidal emf (for example, one can

Fig. 1. Efficiency of brushless motors:
a — with independent connection of the windings: 1 — ideal case; 2 — \(k=0.75,\ \tau_0=1.0\); 3 — \(k=0.75,\ \tau_0=0.5\); 4 — \(k=0.75,\ \tau_0=0.2\); 5 — \(k=0.75,\ \tau_0=0.6\); 6 — \(k=1.0,\ \tau_0=0\).
b — with the phases connected in “star” or “delta”: 1 — ideal case; 2 — \(k=\frac{5}{6},\ \tau_0=1.0\); 3 — \(k=\frac{5}{6},\ \tau_0=0\); 4 — \(k=\frac{5}{6},\ \tau_0=0.5\); 5 — \(k=\frac{5}{6},\ \tau_0=0\); 6 — \(k=1.0,\ \tau_0=0\).

Fig. 2. Effect of relative inductance on efficiency:
1 — for maximum efficiency; 2 — efficiency for maximum useful power; (— — — independent phase switching, – – – phase connection in “star” or “delta”)

Fig. 3. Effect of relative inductance on motor utilization:
1 — for the speed corresponding to maximum efficiency; 2 — for the speed corresponding to maximum useful power

to analyze the independent switching-on of concentrated phase windings with a diametral pitch under a rectangular excitation field). Regardless of the results of these studies, one may conclude that a sinusoidal EMF waveform, when using the available means of approximating the phase-current waveform to a sinusoidal one, is one of the best, ensuring high performance indices for brushless motors.

REFERENCES

1. Yu. N. Kroneberg, E. R. Geints. The torque equation of brushless DC motors with a nonsinusoidal excitation field. (Article in the present collection.)