C ALCULATION OF TORQUE USING THE FINITE ELEMENT PROCEDURE

(a) Simple, fast procedure for sinewound machines with low saturation level

Fig. 2.24 shows the general cross-section of an interior-magnet brushless AC motor, and Fig. 2.25 shows the reference axes of phases a, b, and c together with the rotor reference axes d and q. These axes are consistent with the equations for variation of phase inductances with rotor position in §2.6. The phasor diagram is reproduced in Fig. 2.63 for a normal motoring condition.

T ' m p[Q_1dI_q ! Q_1qI_d] _(2.113) Q_1d ' kw1Tph 2 B_1dDL_stk p ; Q1q ' k_w1T_ph 2 B_1qDL_stk p (2.114) V ' E % jX_dI_d % jX_qI_q % R_phI _(2.112)

I_d ' I cos $ ' ! I sin ( and I_q ' I sin $ ' I cos (. _(2.111)

The current phasor is defined by its r.m.s. value I and its phase angle ( relative to the open-circuit EMF E. The phasor values are I = Id + jIq and E = 0 + jEq = jTQ1Md, where Q1Md is the r.m.s. fundamental

flux-linkage per phase on open-circuit, produced by the magnet: see eqn. (2.57).

The angle ( in the phasor diagram is the angle between the current phasor I and the q-axis, and is measured positive if I leads E. In Fig. 2.63, ( < 0. The angle $ is the angle between the current phasor and the d-axis, such that $ = B/2 + (. Thus

In the physical cross-section of the machine, $ is the angle between the axis of the stator MMF distribution and the d-axis. When $ = 0, the stator MMF is aligned with the d-axis in the magnetizing direction, and Id = I with Iq = 0. When $ = 180E, Id = !I with Iq = 0, and the stator MMF is aligned with

the negative d-axis in the demagnetizing direction. When $ = 90E and ( = 0, the current is "in the q- axis": I_q = I with I_d = 0. In surface-magnet sinewave motors at low speed, the current is normally oriented in the q-axis with ( = 0.

The phasor voltage V = Vd + jVq is the airgap voltage produced by the r.m.s. fundamental airgap flux-

linkage QQQQ₁ under load: thus V = jTQQQQ₁ = Q1d + jQ1q. The phasor diagram is completed by the equation

where I_d is written as the phasor I_d = Id + j0 and Iq is written as the phasor Iq = 0 + jIq. Ra is the phase

resistance. In this equation, both Xd and Xq should incorporate the slot, end-turn, and differential

leakage components (and any skew-leakage component) in addition to the main airgap component associated with the fundamental MMF distribution of the stator winding.

In a sinewound motor we have seen in §2.8 that the torque is computed directly from the fundamental flux-linkage per phase QQQQ₁ and the phase current I using the equation

where p is the number of pole-pairs and m is the number of phases, and Q1d and Q1q are the components

of QQQQ₁ resolved along the d- and q-axes. This is the average torque produced by the fundamental harmonic components of the airgap flux and the stator MMF distribution. It does not include cogging torque or any harmonic torques. However, provided that the correct fundamental components Id, Iq,

Q_1d and Q1q are computed, it is the correct running torque for a sinewave motor, and in a well-designed

motor with a sinusoidal back EMF, it is virtually constant.

Eqn. (2.113) does not require the d-axis flux-linkage to be resolved into the separate components contributed by the magnet and the d-axis armature current; neither does it even require a knowledge of Xd or Xq: these are necessary only when the current has to be computed from the voltage equation

(2.64) or (2.110). If the current is known in magnitude and direction, as is normally the case with a current-regulated drive, then Xd and Xq are not required.

The fundamental flux-linkages Q1d and Q1q are obtained from the finite-element procedure as follows.

First, the d-axis is aligned with the axis of phase a. Then the field solution is obtained for a range of values of I and (, where I = Id + jIq = I ej(B/2 + (), from ( = !90E to +90E (i.e., $ ranges from 0 to 180E). For

each solution the airgap flux-density waveform is Fourier-analyzed to give B1d and B1q, where B1d is

the peak value of the d-axis component and B1q is the peak value of the q-axis component. Then Qd1 and

Q_q1 are obtained using the same form as eqn. (2.57), i.e.

The leakage components I_dL_F and I_qL_F should be added to Q_1d and Q_1q respectively. If the leakage reactance is saturable, the flux-MMF diagram technique should be used instead (see below). These values are substituted in eqn. (2.113) to produce a set of torque curves as illustrated in Fig. 2.64.

I_k ' C[a,k] i_a % C[b,k] i_b % C[c,k] i_c. _(2.115)

i_a ' I_sp cos $

i_b ' I_sp cos ($ ! 2B/3)

i_c ' I_sp cos ($ % 2B/3)

(2.118)

Fig. 2.65 Direction of currents and stator MMF around a S magnet for three values of (.

i_a ' I_d cos 2 ! I_q sin 2 i_b ' I_d cos (2 ! 2B/3) ! I_q sin (2 ! 2B/3) i_c ' I_d cos (2 % 2B/3) ! I_q sin (2 % 2B/3) (2.116) i_a ' I_d i_b ' I_d cos (!2B/3) ! I_q sin (!2B/3) ' (!I_d % 3I_q)/2 i_c ' I_d cos (2B/3) ! I_q sin (2B/3) ' (!I_d ! 3I_q)/2. (2.117) Fig. 2.64 Torque vs. torque angle (, for various currents The values of currents in the individual slots

are obtained from the conductor location

vector (CLV), that is, an array C which has m columns, one for each phase, and n rows, where n is the number of slots. The value of the (j,k)th element is the number of conductors belonging to the jth phase in the kth slot. C[j,k] is signed according to the direction of the conductors. Then for the kth slot the total current is

Instantaneous phase currents ia, ib and ic are

obtained from the phasor values by Park’s transformation:

where I_d and I_q are peak values (i.e., the values taken from the phasor diagram and multiplied by /2). Since the d-axis is aligned with phase a, 2 = 0 and eqns. (2.116) simplify to

Substituting from eqn. (2.111), with $ = ( + 90E the angle between the current and the d-axis,

where Isp = /2*I* is the setpoint current of the current-regulator. In PC-BDC the sign conventions are

such that the main magnet pole analyzed in the finite-element model is a south pole. Fig. 2.65 shows the directions of the current for three different values of (: ( = 0 is the normal "q-axis" control, ( = !90E is "magnetizing" with $ = 0, and +90E is demagnetizing with $ = 180E.

It is not always necessary to compute the entire set of T((,I) curves. If the current is regulated to be sinusoidal, both I and ( are controlled and only one finite-element computation is needed at that point.

Fig. 2.66 i-psi loop

Fig. 2.67 i-psi loop

T_e ' m

2B × W. (2.119)

(b) Flux-MMF diagram technique

Method (a) is fast because it uses a fixed mesh and simple post-processing of the finite-element field solution, but it fails to give adequate results if the machine is not sinewound, or if the airgap flux- distribution is not sinusoidal, or if there is significant saturation.

Moreover. under saturated conditions the values of E and Xd become indeterminate, because the

superposition implied by the equation Q_d = Q_1Md +

LdId, (or the equivalent voltage equation Vq = Eq + XdId)

no longer applies, so that it is impossible to resolve Qd

(or Vq) uniquely into separate components, one

attributed to the magnet and the other to the current.

Probably the most realistic physical interpretation is that both Eq and Xd are reduced by saturation.

In the q-axis, the question of uniqueness does not appear to arise, because the linear theory of sinewound machines tacitly assumes that there is no Ed produced by "magnet flux" in the q-axis. But

under saturated conditions a component of E_d can appear, while X_q is certainly reduced by saturation, often to a severe degree.

Under these circumstances the most powerful method for computing the torque is to calculate a sequence of finite-element field solutions at intervals of rotation throughout an electrical cycle. The current (in all phases) set to the correct value for each rotor position, and the i-R loop is plotted, that is, the loop formed by a point whose coordinates are the phase current i and the phase flux-linkage R. With a sinewound machine having ideal sinusoidal current waveforms, the i-R loop or "energy- conversion loop" is elliptical, its major axis being oriented at an angle that depends on the phase angle between the current and the d-axis. The average electromagnetic torque is proportional to the area W of the i-R loop: thus

The phase flux-linkage R is extracted from the finite-element solution. The flux through a single coil is N = L_stk (A_r ! A_g) [Wb], where A_r is the vector potential at the "return" coilside and A_g is the vector potential at the "go" coilside, and Lstk is the stack length. The flux-linkage is R = TcN [V-s], where Tc

is the number of turns in the coil.

If required, the d! and q-axis components of flux-linkage can be approximated using the transformation, Rd = 2 Ra/3 ! (Rb + Rc)/3 and Rq = (Rb ! Rc)//3. However, this transformation is not

of interest unless the winding is sinewound.

The flux-MMF diagram technique works for any machine with any drive. It is not restricted to sinewound machines or sinewave drives, and works equally well with machines having non-sinusoidal EMF waveforms and squarewave drive, regardless of the level of saturation.

The main limitation of the method is that the phase currents must be known at every step of rotation, because the finite-element computation is "current- driven".

Fig. 2.67 shows an example of the i-R loop computed for a "squarewave" brushless DC motor, with the current regulated to follow a 120E squarewave by chopping.

26_{Eqn. (2.120) therefore applies to surface-magnet motors but is incomplete for embedded-magnet motors.} T_avg ' 2B 0 Te(2)d2 (2.121) T_eT_m ' e₁i₁ % e₂i₂ % e₃i₃ % ... % e_mi_{m (2.120)} e₁ ' e_pk cos Tt i₁ ' i_pk cos (Tt ! () e₂ ' e_pk cos (Tt ! 2B/3) i₁ ' i_pk cos (Tt ! 2B/3 ! () e₃ ' e_pk cos (Tt % 2B/3) i₁ ' i_pk cos (Tt % 2B/3 ! () (2.122) T_e ' 3 E I T_m cos ( (2.123) e₁i₁ ' e_pki_pk cos Tt cos (Tt ! () ' epkipk 2 [cos ( % cos (2Tt ! ()] (2.124) 2.16 TORQUE AND TORQUE RIPPLE

Two main sources of torque ripple can be identified in brushless permanent-magnet motors: (1) electromagnetic torque ripple; and

(2) cogging torque.

Electromagnetic torque in brushless PM machines is the result of the interaction between the phase currents i1, i2,... and the EMF’S e1, e2,... generated by the rotation of the magnets. Ideally the energy

conversion is described by the equation

where Tm is the speed in rad/sec and m is the number of phases. It is important to recognize Te as the

instantaneous torque and not the average torque. The average torque (averaged over one or more revolutions) is the main output parameter,

and in most cases Tavg is the parameter that is quoted, with never any mention of the instantaneous

torque. Quite possibly the credit for this must go to the designers of classical DC and AC machines who, for the first century of the history of electric machines, delivered such smooth torque that no-one gave a second thought to the existence of torque ripple, and the whole subject of instantaneous torque and the theory of electromechanical energy conversion was academic.

Now in the age of electronic drives, torque ripple is an important topic, probably for three reasons: (a) power electronics has opened up many applications which are extremely sensitive to torque

ripple, e.g., automotive power steering, machine tool feed drives, and computer disk drives; (b) power electronics has made it possible to use motors which do not inherently deliver smooth

torque, except in their most ideal theoretical forms; and

(c) power electronics and associated digital controls can exacerbate the torque ripple problem. It is worth reviewing the principles of smooth torque production starting from eqn. (2.120), even though this equation includes only the “alignment” torque of the permanent magnets and does not include any reluctance torque.26 _{In AC machines the most classical instance of “constant torque” is the three-} phase machine wherein the EMF’s and currents are balanced polyphase quantities:

Substituting in eqn. (2.120), and recognizing the r.m.s. values E = epk//2 and I = ipk//2, we get

T_e ' m T_m[EqIq % (Xd ! Xq)IdIq] (2.127) T_rel ' m T_m × 2Bf[Ld ! Lq] idiq ' mp 2 [(Ld ! Lq)idiq] (2.128) i_d ' 2

3[ia cos 2 % ib cos (2 ! 2B/3) % ic cos (2 % 2B/3)];

i_q ' ! 2

3[ia sin 2 % ib sin (2 ! 2B/3) % ic sin (2 % 2B/3)].

(2.129)

e₁ ' e_pk cos Tt i₁ ' i_pk cos (Tt ! ()

e₂ ' e_pk cos (Tt ! B/2) i₁ ' i_pk cos (Tt ! B/2 ! () (2.125)

T_e ' 2 E I

T_m cos (. (2.126)

which has an average value (EI/2) cos ( with a double-frequency oscillatory component whose amplitude is greater than the average value unless ( = 0 (i.e., unless e1 and i1 are in phase). This

enormous torque ripple is the reason why pure single-phase AC motors are used in only a limited number of applications. It also explains the use of auxiliary capacitors with split-phase induction motors operating from a single-phase supply. With two phases the phase displacement between phases is ideally B/2 or 90E instead of the 2B/3 or 120E used in three-phase machines, and if

then the sum e1i1 + e2i2 yields

In balanced polyphase systems it is always the case that as the torque contribution of one phase varies, the variation is exactly compensated by variations in the other phases.

Eqn. (2.120) also suggests the other main principle of attaining constant instantaneous torque, which is to maintain one or two of the product terms constant for a fixed angle of rotation, and then “commutate” to another product term or pair of terms. For example, Fig. 2.7 shows how this is accomplished in the 3-phase brushless motor. The angle of rotation over which one of the product terms (such as T1) can be kept constant is limited by the magnet arc and the winding distribution, and

although the current can be held constant for very nearly 180E of rotation, it is not possible to maintain a flat-topped phase EMF waveform wider than about 175E or so. For this reason it is impossible to achieve constant torque in a two-phase brushless motor, and it is better to settle for three phases and 120E flat tops, which can be adjusted to minimize any ripple arising at the commutation points. Even with a motor which has ideal sinusoidal or flat-topped trapezoidal EMF waveforms, the current waveform may depart from the ideal sinusoid or 120E squarewave because of chopping (PWM) and commutation in the drive. Moreover, at high speeds the ideal sine or 120E squarewave current cannot be achieved because of the combination of series inductance and the growth in the EMF relative to the available supply voltage.

In sinewound motors with sinusoidal supply currents the torque equation has already been expressed in various forms such as eqns. (2.60), (2.62), (2.69), (2.78), and (2.113): all of these equations refer to the average torque. Another variant of these equations is

The reluctance torque is identifiable as the second term in this equation, and this is the average value since E_q, I_q and I_d are all r.m.s. quantities. However, the instantaneous reluctance torque Trel can be determined if the instantaneous currents are used instead of the r.m.s. values:

where T_m = 2B f /p and i_d and i_q are the instantaneous d- and q-axis currents, obtained from the instantaneous phase currents by Park's transformation:

and 2 is the angle between the d-axis and the axis of phase a. The inverse Park transformation is given by eqn. (2.116). This is an interesting illustration of the fact that Park’s transformation is valid under transient conditions, or with nonsinusoidal currents, provided only that the windings are sinewound.

Fig. 2.68 Representation of magnet by equivalent air-cored coil

B ' B_r % µ_recµ₀H _(2.131)

W_f ' 1

2 m BH dV (2.130)

Fig. 2.69 Result of a small perturbation in magnet operating point from X to Y.