Torque Equation of Brushless DC Motors with a Nonsinusoidal Excitation Field
Abstract Generated abstract
This paper develops torque and power equations for brushless DC motors supplied by semiconductor commutators when the excitation field and phase voltages contain nonsinusoidal harmonics. Using the general theory of electrical machines, it represents phase voltages, currents, and flux linkages as harmonic vector components, transforms the stator voltage equations into rotating coordinates, and separates forward, reverse, and zero-sequence harmonic effects. The resulting formulas describe constant rotating torque, pulsating torque, and consumed active power for a general multiphase machine, with a detailed specialization to three-phase motors. The analysis shows how higher time and space harmonics, including zero-sequence components, can contribute to driving or pulsating torques and provides equations useful for assessing torque characteristics, motion uniformity, and rotor field-shape design.
Full Text
TORQUE EQUATION OF BRUSHLESS DC MOTORS WITH A NON-SINUSOIDAL EXCITATION FIELD
Yu. N. Kroneberg, E. R. Geints
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
Supplying brushless DC motors from semiconductor commutators, with a non-sinusoidal form of the excitation field, leads to the appearance, in addition to the fundamental one, of a group of additional rotating and pulsating torques. Their occurrence is due to the higher time and space components of the magnetic fields of the stator and rotor interacting in the air gap of the machine. In studying such motors, it is more convenient to use the methods of the general theory of electrical machines, which make it possible to carry out a direct assessment of the influence of one or another spectrum of harmonics on the characteristics of the machine.
The phase voltages of brushless motors (Fig. 1) can be represented in the form of the following harmonic series:
\[ u_{\phi}^{(1)}=\frac{4U}{\pi}\sum_{n=1}^{\infty} \frac{\sin(2n-1)\frac{\pi}{2}\,k}{2n-1}\cos(2n-1)\omega t, \tag{1} \]
\[ u_{\phi}^{(\Delta)}=\frac{4U}{\pi}\sum_{n=1}^{\infty} \frac{\sin(2n-1)\frac{\pi}{3}-\sin(2n-1)\frac{\pi}{2}\sin(2n-1)\frac{\pi}{2}\,k}{2n-1} \times \cos(2n-1)\omega t, \tag{2} \]
\[ u_{\phi}^{(\lambda)}=\frac{4U}{3\pi}\sum_{n=1}^{\infty} \frac{\sin(2n-1)\frac{\pi}{2}\,k \left[1+\sin(2n-1)\frac{\pi}{6}-\sin(2n-1)\frac{\pi}{2}\right]}{2n-1} \times \cos(2n-1)\omega t, \tag{3} \]
where the duty factor of the switching-on of the power elements of the commutator (the ratio of the duration of switching-on to the half-period) for the circuits under consideration usually lies within the limits \(k=\frac{2}{3}\div 1.0\). Using (1)–(3), the generalized expression for the instantaneous value of the phase voltage of an \(m\)-phase machine is
\[ u_{\phi}=u_{\phi 1}+\sum_{n=1}^{\infty}u_{\phi(2mn+1)} +\sum_{n=1}^{\infty}u_{\phi(2mn-1)} +\sum_{n=1}^{\infty}u_{\phi[m(2n-1)]}. \tag{4} \]
Fig. 1. Winding connection diagrams and phase voltages of motors:
a — independent connection; b — phase connection in “delta”; c — phase connection in “star”; \(U\) — voltage of the direct-current source; \(\omega t\) — rotor angle of rotation; \(K\) — commutator.
Here the index “1” denotes the fundamental harmonic; the index \((2mn \pm 1)\) denotes harmonics producing rotating fields in the forward and reverse directions; \([m(2n-1)]\) denotes harmonics forming the zero axis.
The voltage components (4), applied to the linear circuits of a machine with active stator-phase resistances \(R_s\), will cause harmonic currents of the same types to flow through the phase windings. Using the general rule for representing an \(m\)-phase system of symmetrical voltages, currents, and flux linkages in the form of resultant vectors [1],
\[ \bar U_{\sigma} = \bar U_{\sigma 1} + \sum_{n=1}^{\infty} \bar U_{\sigma(2mn+1)} + \sum_{n=1}^{\infty} \hat U_{\sigma(2mn-1)}, \]
\[ \bar i_{\sigma} = \bar i_{\sigma 1} + \sum_{n=1}^{\infty} \bar i_{\sigma(2mn+1)} + \sum_{n=1}^{\infty} \hat i_{\sigma(2mn-1)}, \tag{5} \]
\[ \bar \Psi_{\sigma} = \bar \Psi_{\sigma 1} + \sum_{n=1}^{\infty} \bar \Psi_{\sigma(2mn+1)} + \sum_{n=1}^{\infty} \hat \Psi_{\sigma(2mn-1)}, \]
the voltage-balance equation of the stator winding in stationary coordinates can be written as
\[ \bar U_{\sigma} = \bar i_{\sigma} R_s + \frac{d}{dt}\,\bar \Psi_{\sigma}. \tag{6} \]
We shall assume that the rotor field contains all odd spatial harmonics. Like-named time harmonics of the stator field that are not multiples of the number of phases, and the spatial harmonics of the rotor field, interacting pairwise, form a group of additional rotating torques, since the rotational speeds of these fields, expressed in electrical angular units, are the same. Consequently, for each of the components of the voltage \(\bar U_\sigma\), the longitudinal and transverse axes of the rotor, rotating at the speed of the harmonic with respect to which equation (6) is solved, may be chosen as the axes of transformation. Then the complete solution can be obtained by the method of superposition. The vectors of equations (5) found in this way admit, when necessary (for determining pulsating torques), a simple transformation from synchronous coordinates to coordinates rotating with some constant slip \([1, 2]\).
On the basis of the foregoing, equation (6), transformed to a rotating coordinate system, has the form:
\[ \bar U_{s1}=\bar i_{s1}R_s+j\omega \bar\Psi_{s1}, \]
\[ \bar U_{s(2mn+1)}=\hat i_{s(2mn+1)}R_s+j(2mn+1)\omega \bar\Psi_{s(2mn+1)}, \tag{7} \]
\[ \bar U_{s(2mn-1)}=\bar i_{s(2mn-1)}R_s-j(2mn-1)\omega \bar\Psi_{s(2mn-1)}. \]
Harmonics that are multiples of the number of phases form single-phase pulsating fields, which interact only with spatial harmonics of the rotor of the same order, creating rotating and pulsating torques. The phase currents of these harmonics can be found by the general method for determining zero-sequence currents from the solution of the equation
\[ u_{\phi[m(2n-1)]}=i_{\phi[m(2n-1)]}R_s+\frac{d}{dt}\Psi_{\phi[m(2n-1)]}. \tag{8} \]
Representing each of the sinusoidal components of equation (8) in the form of the real parts of the corresponding rotating vectors \([1]\) and taking into account that, in the steady state,
\[ \frac{d}{dt}\Psi_{\phi[m(2n-1)]} = \operatorname{Re}\left\{jm(2n-1)\omega \bar\Psi_{s[m(2n-1)]}\right\}, \]
we obtain
\[ \bar U_{s[m(2n-1)]} = \bar i_{s[m(2n-1)]}R_s + jm(2n-1)\omega \bar\Psi_{s[m(2n-1)]}. \tag{9} \]
To find the electromagnetic torque we shall use the general expression
\[ M=\frac{m}{2}\operatorname{Re}\left[j\bar\Psi_s\hat i_s\right]. \]
After substituting the components of the resultant flux-linkage and current vectors,
\[ \bar\Psi_s = \bar\Psi_{s1} + \sum_{n=1}^{\infty}\bar\Psi_{s(2mn+1)} + \sum_{n=1}^{\infty}\hat\Psi_{s(2mn-1)}, \]
\[ \bar i_s = \bar i_{s1} + \sum_{n=1}^{\infty}\bar i_{s(2mn+1)} + \sum_{n=1}^{\infty}\hat i_{s(2mn-1)}, \tag{10} \]
and also neglecting, because of their smallness, the alternating torques from the interaction of the higher harmonic fields with one another, we obtain
\[ \begin{aligned} M_{1,(2mn\pm 1)}={}&\frac{m}{2}\operatorname{Re}j\left\{ \overline{\Psi}_{s1}\hat{i}_{s1} +\sum_{n=1}^{\infty}\overline{\Psi}_{s(2mn+1)}\hat{i}_{s(2mn+1)} +\sum_{n=1}^{\infty}\hat{\Psi}_{s(2mn-1)}\overline{i}_{s(2mn-1)} \right\} \\ &+\frac{m}{2}\operatorname{Re}j\left\{ \sum_{n=1}^{\infty}\overline{\Psi}_{s1}\hat{i}_{s(2mn+1)} +\sum_{n=1}^{\infty}\overline{\Psi}_{s1}\overline{i}_{s(2mn-1)} +\sum_{n=1}^{\infty}\overline{\Psi}_{s(2mn+1)}\hat{i}_{s1} +\sum_{n=1}^{\infty}\hat{\Psi}_{s(2mn-1)}\hat{i}_{s1} \right\}. \end{aligned} \tag{11} \]
The first term of the equation represents the constant torque produced by the pairwise interaction of time and space harmonics of one and the same order; the second is the sum of pulsating torques.
Fig. 2. System of phase (ABC) and rotating coordinates (dq)
The additional electromagnetic torque produced by the field of zero-sequence components [3], without taking into account pulsating torques of higher orders,
\[ M_{1m(2n-1)}=\sum_{n=1}^{\infty} \left\{ -mM_{M[m(2n-1)]}i_f\operatorname{Re}\left[\overline{i}_{s[m(2n-1)]}\right] \sin [m(2n-1)]\ldots \right\}, \tag{12} \]
where
\[ \gamma=(\omega t-\gamma_0) \]
is the misalignment angle between the longitudinal axis of the rotor and the magnetic axis of phase \(A\) (Fig. 2);
\(\gamma_0\) is the initial angular position of the rotor.
The useful torque on the motor shaft is equal to the algebraic sum of the individual components (11) and (12). In carrying out the summation it is necessary to take into account not only the direction of rotation of the magnetic fields of the time and space harmonics, but also the mutual relation of their phases. Therefore, despite the fact that harmonics of order \((2mn-1)\) form an oppositely directed field, in a number of cases their interaction leads to the appearance of a positive driving torque, for example under the conditions adopted by the author in [4] (rectangular waveform of the emf and of the phase current).
The instantaneous value of the consumed active power
\[ p_{1\Sigma}=\frac{m}{2}\operatorname{Re}[\overline{U}_s\hat{i}_s] +m\sum_{n=1}^{\infty}\operatorname{Re}[\overline{U}_{s[m(2n-1)]}] \operatorname{Re}(\hat{i}_{s[m(2n-1)]}) \tag{13} \]
is composed of the sum of the powers of the individual harmonics and the excitation power of the zero axis.
Let us find the solution of equations (12) and (13) for three-phase motors, making it possible to reveal the role of the harmonic zero-sequence component. In the absence of asymmetry of the magnetic circuit, the coefficients of self- and mutual inductance of the phase windings practically do not depend on the rotor position,
\[ \begin{gathered} L_a=L_b=L_c=l_0,\\ M_{ab}=M_{bc}=M_{ca}=m_0, \end{gathered} \tag{14} \]
and the coupling coefficients of the stator winding phases with the polyharmonic field of the rotor form trigonometric series of the form:
\[ \begin{aligned} M_{af}={}&M_{M1}\cos\gamma+\sum_{n=1}^{\infty}M_{M(2mn+1)}\cos(2mn+1)\gamma+\\ &+\sum_{n=1}^{\infty}M_{M(2mn-1)}\cos(2mn-1)\gamma +\sum_{n=1}^{\infty}M_{M[m(2n-1)]}\cos[m(2n-1)]\gamma,\\ M_{bf}={}&M_{M1}\cos(\gamma-120^\circ) +\sum_{n=1}^{\infty}M_{M(2mn+1)}\cos(2mn+1)(\gamma-120^\circ)+\\ &+\sum_{n=1}^{\infty}M_{M(2mn-1)}\cos(2mn-1)(\gamma-120^\circ)+\\ &+\sum_{n=1}^{\infty}M_{M[m(2n-1)]}\cos\{m(2n-1)\}(\gamma-120^\circ), \end{aligned} \tag{15} \]
\[ \begin{aligned} M_{cf}={}&M_{M1}\cos(\gamma+120^\circ) +\sum_{n=1}^{\infty}M_{M(2mn+1)}\cos(2mn+1)(\gamma+120^\circ)+\\ &+\sum_{n=1}^{\infty}M_{M(2mn-1)}\cos(2mn-1)(\gamma+120^\circ)+\\ &+\sum_{n=1}^{\infty}M_{M[m(2n-1)]}\cos[m(2n-1)](\gamma+120^\circ). \end{aligned} \]
Taking into account equations (14) and (15), the resultant flux-linkage vectors in synchronous coordinates are
\[ \overline{\Psi}_{s1}=\overline{\Psi}_{\sigma1}e^{-j\gamma} =L_s\overline{i}_{s1}+M_{M1}i_f, \]
\[ \overline{\Psi}_{s(2mn+1)} =\overline{\Psi}_{\sigma(2mn+1)}e^{-j(2mn+1)\gamma} =L_s\overline{i}_{s(2mn+1)}+M_{M(2mn+1)}i_f, \]
\[ \overline{\Psi}_{s(2mn-1)} =\overline{\Psi}_{\sigma(2mn-1)}e^{-j(2mn-1)\gamma} =L_s\overline{i}_{s(2mn-1)}+M_{M(2mn-1)}i_f, \]
\[ \overline{\Psi}_{s[m(2n-1)]} =L_0\overline{i}_{s[m(2n-1)]} +M_{M[m(2n-1)]}i_f e^{j[m(2n-1)]\gamma} \tag{16} \]
and, according to equations (7) and (9), the required currents are
\[ \overline{i}_{s1} =\frac{U_{M1}e^{j\gamma_0}}{[R_s+jX_a]} \left\{1-j\left|\frac{E_{M1}}{U_{M1}}\right|e^{-j\gamma_0}\right\}, \]
\[ \overline{i}_{s(2mn+1)} =\frac{U_{M(2mn+1)}e^{j(2mn+1)\gamma_0}} {[R_s+j(2mn+1)X_a]} \left\{1-j\left|\frac{E_{M(2mn+1)}}{U_{M(2mn+1)}}\right| e^{-j(2mn+1)\gamma_0}\right\}, \]
\[ \bar{i}_{s(2mn-1)}= \frac{U_{M(2mn-1)}e^{j(2mn-1)\omega t}} {\left[R_s+j(2mn-1)X_a\right]} \left\{1-j\left|\frac{E_{M(2mn-1)}}{U_{M(2mn-1)}}\right| e^{-j(2mn-1)\gamma_0}\right\}, \tag{17} \]
\[ \bar{i}_{s[m(2n-1)]}= \frac{U_{M[m(2n-1)]}e^{j[m(2n-1)]\omega t}} {\left[R_s+jm(2n-1)X_0\right]} \left\{1-j\left|\frac{E_{M[m(2n-1)]}}{U_{M[m(2n-1)]}}\right| e^{-j[m(2n-1)]\gamma_0}\right\}, \]
where in equations (15), (16), and (17)
\[
L_s=(l_0-m_0),\quad X_a=\omega L_s
\]
— the inductance of the phase winding and the reactance of the armature reaction, taking into account the influence of the other phases;
\[
L_0=(l_0+2m_0),\quad X_0=\omega L_0
\]
— the inductance and reactance of the phase winding for zero-sequence currents;
\[
M_{M1},\ M_{M(2mn\pm1)},\ M_{M[m(2n-1)]}
\]
— amplitudes of the mutual-induction coefficients of the magnetic fields of the corresponding harmonics;
\[
|U_{M1}|,\ |E_{M1}|=\omega M_{M1}i_f
\]
— absolute values of the amplitudes of the phase voltage and the phase emf of the fundamental harmonic;
\[
|U_{M(2mn\pm1)}|,\ |E_{M(2mn\pm1)}|,\ |U_{M[m(2n-1)]}|,\ |E_{M[m(2n-1)]}|
\]
— the same for the higher harmonics.
The simultaneous solution of (11)–(13), (16), and (17) determines the rotating electromagnetic torque
\[ \begin{aligned} M_{\mathrm{rot.}}=M_p\Bigg\{& \frac{R_s^2}{\left[R_s^2+X_a^2\right]} \left[\sin\gamma_0-\frac{X_a}{R_s}\cos\gamma_0- \left|\frac{E_{M1}}{U_{M1}}\right|\right] \\ &+\sum_{n=1}^{\infty} \frac{E_{M(2mn+1)}U_{M(2mn+1)}R_s^2} {E_{M1}U_{M1}\left[R_s^2+(2mn+1)^2X_a^2\right]} \left[(-1)^n\sin(2mn+1)\gamma_0 \right.\\ &\left.\qquad\qquad -(2mn+1)\frac{X_a}{R_s}\cos(2mn+1)\gamma_0 -\left|\frac{E_{M(2mn+1)}}{U_{M(2mn+1)}}\right|\right] \\ &+\sum_{n=1}^{\infty} \frac{E_{M(2mn-1)}U_{M(2mn-1)}R_s^2} {E_{M1}U_{M1}\left[R_s^2+(2mn-1)^2X_a^2\right]} \times \left[(-1)^{(n-1)}\sin(2mn-1)\gamma_0 \right.\\ &\left.\qquad\qquad -(2mn-1)\frac{X_a}{R_s}\cos(2mn-1)\gamma_0 -\left|\frac{E_{M(2mn-1)}}{U_{M(2mn-1)}}\right|\right] \\ &+\sum_{n=1}^{\infty} \frac{E_{M[m(2n-1)]}U_{M[m(2n-1)]}R_s^2} {E_{M1}U_{M1}\left[R_s^2+m^2(2n-1)^2X_0^2\right]} \left[(-1)^n\sin[m(2n-1)]\gamma_0 \right.\\ &\left.\qquad\qquad -m(2n-1)\frac{X_0}{R_s}\cos[m(2n-1)]\gamma_0 -\left|\frac{E_{M[m(2n-1)]}}{U_{M[m(2n-1)]}}\right|\right]\Bigg\} \end{aligned} \tag{18} \]
and the active power consumed
\[ P_{1\,\mathrm{cons.}}=P_{1P}\left\{ \frac{R_s^2}{\left[R_s^2+X_a^2\right]} \left[1-\left|\frac{E_{M1}}{U_{M1}}\right| \left[\sin\gamma_0+\frac{X_a}{R_s}\cos\gamma_0\right]\right] +\right. \]
\[ \left. +\sum_{n=1}^{\infty} \frac{U_{M(2mn+1)}^2R_s^2} {U_{M1}^2\left[R_s^2+(2mn+1)^2X_a^2\right]} \left[1-\left|\frac{E_{M(2mn+1)}}{U_{M(2mn+1)}}\right| \left[(-1)^n\sin(2mn+1)\gamma_0+\right.\right.\right. \]
\[ \begin{aligned} &+ (2mn+1)\frac{X_a}{R_s}\cos(2mn+1)\gamma_0\Bigg]\Bigg] + \sum_{n=1}^{\infty} \frac{U^2_{M(2mn-1)}R_s^2} {U^2_{M1}\left[R_s^2+(2mn-1)^2X_a^2\right]} \times \\[3pt] &\times \Bigg[1-\left|\frac{E_{M(2mn-1)}}{U_{M(2mn-1)}}\right| \Big[(-1)^{(n-1)}\sin(2mn-1)\gamma_0 + \\[3pt] &\qquad\qquad + (2mn-1)\frac{X_a}{R_s}\cos(2mn-1)\gamma_0\Big]\Bigg] + \sum_{n=1}^{\infty} \frac{U^2_{M1[m(2n-1)]}R_s^2} {U^2_{M1}\left[R_s^2+m^2(2n-1)^2X_0^2\right]} \times \\[3pt] &\times \Bigg[1-\left|\frac{E_{M1[m(2n-1)]}}{U_{M1[m(2n-1)]}}\right| \Big[(-1)^n\sin[m(2n-1)]\gamma_0 + \\[3pt] &\qquad\qquad + m(2n-1)\frac{X_0}{R_s}\cos[m(2n-1)]\gamma_0\Big]\Bigg]\Bigg\}, \end{aligned} \tag{19} \]
where
\[ M_{\mathrm{p}}=\frac{mE_{M1}U_{M1}}{2\Omega R_s} \]
is the starting torque of the motor due to the first harmonics;
\[ P_{11}=\frac{m}{2}\frac{U_{M1}^2}{R_s} \]
is the starting power consumed when the motor is supplied by the fundamental voltage harmonic (1)–(3).
The second component of equation (11) combines a group of pulsating torques due to the interaction of unlike time and space harmonics. Taking into account that the magnetic fields of these harmonics move relative to the field of the fundamental harmonic with a certain constant slip \(s=\pm 2mn\), and neglecting, because of their smallness, the additional pulsating torques arising from the interaction of unlike harmonics that form a zero axis, that is, the alternating component of equation (12), we find
\[ \begin{aligned} M_{\mathrm{puls.}}=M_{\mathrm{p}}\Bigg\{& \sum_{n=1}^{\infty} \frac{U_{M(2mn+1)}R_s^2} {U_{M1}\left[R_s^2+(2mn+1)^2X_a^2\right]} \Bigg[ \sin(2mn\omega t+\gamma_0) \\ &\quad -(2mn+1)\frac{X_a}{R_s}\cos(2mn\omega t+\gamma_0) -\left|\frac{E_{M(2mn+1)}}{U_{M(2mn+1)}}\right| \Big[ \cos 2mn(\omega t-\gamma_0) \\ &\quad +(2mn+1)\frac{X_a}{R_s}\sin 2mn(\omega t-\gamma_0) \Big]\Bigg] - \sum_{n=1}^{\infty} \frac{U_{M(2mn-1)}R_s^2} {U_{M1}\left[R_s^2+(2mn-1)^2X_a^2\right]} \times \\ &\times \Bigg[ \sin(2mn\omega t-\gamma_0) -(2mn-1)\frac{X_a}{R_s}\cos(2mn\omega t-\gamma_0) \\ &\quad -\left|\frac{E_{M(2mn-1)}}{U_{M(2mn-1)}}\right| \Big[ \cos 2mn(\omega t-\gamma_0) +(2mn-1)\frac{X_a}{R_s}\sin 2mn(\omega t-\gamma_0) \Big]\Bigg] \\ &\quad + \sum_{n=1}^{\infty} \left[ \frac{E_{M(2mn+1)}}{E_{M1}} + \frac{E_{M(2mn-1)}}{E_{M1}} \right] \frac{R_s^2}{R_s^2+X_a^2} \left[ \sin\gamma_0 -\frac{X_a}{R_s}\cos\gamma_0 -\left|\frac{E_{M1}}{U_{M1}}\right| \right] \Bigg\}. \end{aligned} \tag{20} \]
The obtained equations (18), (19), and (20) can be used to analyze the operation of a brushless motor in steady-state regimes, making it possible, in particular, to determine the form of the angular characteristic of the torque and the uniformity of motion. In addition, they provide a basis for designing a rotor with a rational field shape. The equations considered, although rather cumbersome in the general case, are substantially simplified when analyzing motors with specific winding schemes and with a small number of rotor harmonics.
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