T HE ORIGIN AND DEVELOPMENT OF AC VECTOR CONTROL FOR BRUSHLESS PM MOTORS

The terms direct axis and quadrature axis refer to the two axes of symmetry of the magnetic field system as defined by the field winding or excitation winding. In the DC machine the excitation winding is on the stator9 _{and therefore the d.q axes are fixed to the stator. In the AC synchronous machine the} excitation winding is on the rotor and therefore the d.q axes are fixed to the rotor, Fig. 2.29. The d-axis is the axis of symmetry centred on one rotor pole. Sometimes it is called the polar axis or field axis. The q-axis is also an axis of symmetry and it is known as the interpolar axis. Because there are two axes of symmetry, the rotor is said to have two-axis symmetry. In electrical terms the d- and q-axes are orthogonal, i.e., separated by 90 “electrical degrees”. Indeed the definition of “electrical degrees” is such that there are 180 electrical degrees between consecutive d-axes.

In the permanent-magnet machine the wound poles are replaced by permanent magnets, Figs. 2.30 and 2.31, but the meaning of the d and q axes is unchanged. The interior-magnet motor in Fig. 2.31 is a

salient-pole machine with different inductive properties along the d- and q-axes, but the surface-magnet

motor in Fig. 2.30 is nonsalient-pole, being rotationally symmetric apart from the magnetization of the magnets and the possibility of slight differences in permeability along the d- and q-axes.

To understand vector control we need first to understand the phasor diagram, Fig. 2.32, which is slightly more complex than the one in Fig. 2.28 in that it is drawn for a salient-pole machine with X_d different from Xq. The voltage drop XsI is replaced by two separate voltage drops XdId and XqIq.

Fig. 2.32 Phasor diagram and flux-linkage vector diagram

In the steady-state the r.m.s. voltage V in each phase is related to its r.m.s. flux-linkage Q by a simple equation V = TQ where T is the frequency in electrical rad/sec. We have seen that the phase angle of this voltage is 90E ahead of the flux-linkage. For example, in Fig. 2.32, V is represented by an arrow (called a phasor) which is 90E ahead of the arrow representing Q. Mathematically, the phasor value of V is the complex number V = V ej(* + B/2), the angle (* + B/2) being the phase angle of V relative to the reference axis (the d-axis). Both V and Q are sinusoidal quantities in time, and both are represented in the diagram by their r.m.s. values, while the continual advance of their phase angles is represented by the rotation of all the phasors at the angular velocity T rad/sec. During this rotation the phase displacement between Q and V remains at 90E.

The phasor diagram in Fig. 2.32 is split into two parts. On the left are the electrical quantities, i.e., voltages and currents. On the right are the corresponding magnetic flux-linkages. The separation into two parts makes the diagram clearer. The flux-linkage part of the diagram is usually omitted, but it can be considered to have a physical reality, in that the flux actually rotates in space, at an angular velocity of T elec. rad/s. The flux-linkages can therefore be interpreted as space vectors (if considered as rotating physically in space); or simply as phasors (if considered as time-varying sinusoidal quantities). We can now examine how the entire phasor diagram is built up. In a permanent-magnet machine there is a flux due to the magnets which links all the windings in turn, and gives rise to the flux-linkage Q1Md

in each phase, even when there is no current flowing. Corresponding to this flux-linkage is the “open- circuit” voltage E, which leads Q1Md in phase by 90E, just as V leads Q by 90E. In the phasor diagram,

the flux Q1Md is along the d-axis, and therefore E is along the q-axis. Note that the dq axes in the phasor

diagram on the left-hand side are really fictional axes defined in terms of the time phasor diagram. However, since the time phasor diagram (of voltages and currents) and the space vector diagram (of flux-linkages) both rotate synchronously in their respective coordinate systems, we tend to blur the distinction and regard the dq axes as being the same for both. In common engineering parlance, everyone takes this for granted and one would be thought pedantic if one continually reiterated the distinction between them.

10_{The above theory is called the two-axis theory, two-reaction theory, or dq-axis theory of the synchronous machine. Historically} it dates back to A. Blondel in the 1890's, but its modern exposition is generally attributed to R.H. Park along with others around 1920!1930, notably Doherty, Nickle, and W.V. Lyon. See Park RH, Two-reaction theory of synchronous machinery; I ! Generalized method of analysis, Transactions IEEE, Vol. 48, pp. 716!730, July 1929. One of the most famous accounts of it is given by Charles Concordia in Synchronous Machines, John Wiley, 1951.

T_e ' mp(Q_dI_q ! Q_qI_d) _(2.60)

I_d ' ! I sin (; I_q ' I cos ( _(2.63)

Q_d ' Q_1Md % L_dI_d; Q_q ' L_qI_q. _(2.61)

T_e ' mp[Q_1MdI cos ( ! I2 _{sin ( cos ( (L}

d ! Lq)] (2.64) T_e ' mp[Q_1MdI_q % I_dI_q(L_d ! L_q)] _(2.62) ( ' sin!1 1 4 ! Q_1Md )Q % Q_1Md )Q 2 % 8 (2.65) I ' I_d % j I_{q (2.59)}

When current flows in the stator windings it creates an additional flux which is easier to analyse if we first “resolve” the current into two components in the phasor diagram: Id along the d axis and Iq along

the q axis. In terms of the complex phasor I, this is written

The flux-linkage produced by I_d is L_dI_d, where L_d is the d-axis synchronous inductance. It is in phase with Id and induces a voltage XdId which is 90E ahead of Id (i.e., parallel to the q axis); then Xd = TLd

is the d-axis synchronous reactance. Likewise the current Iq produces a flux-linkage LqIq and a voltage

X_qI_q which is parallel to the negative d axis, with X_q = TL_q the q-axis synchronous reactance. The total voltage at the phase terminals is the sum of the component voltages E, XdId, and XqIq, added together

“vectorially” by means of the polygon formed by the respective phasors placed nose-to-tail. Similarly the total flux-linkage is the vector sum of the component flux-linkages Q1Md, LdId and LqIq.10

The phasor diagram is useful in understanding how the torque is limited by the voltage and current available from the drive. Neglecting losses, the electromagnetic torque is given by

where m is the number of phases, p is the number of pole-pairs, and Qd and Qq are the d- and q- axis

components of the r.m.s. flux-linkage per phase. In the d-axis the flux-linkage has two components, Q_1Md due to the magnet and LdId due to the stator current component Id. In the q-axis there is no

magnet flux but only the armature-reaction component LqIq. Thus

If we substitute the expressions for Qd and Qq in eqn. (2.60), we get

which shows that there are two components of torque, a permanent-magnet alignment torque Q_1MdI_q

and a reluctance torque IdIq(Ld ! Lq). If there is no saliency Ld = Lq and no reluctance torque. If the

magnet flux is constant the torque is proportional to Iq, and the torque constant kT = Te/I is constant.

The controller should then maintain I = Iq, which is sometimes called “quadrature control”. If,

however, there is saliency and L_d and L_q are unequal, the mix of permanent-magnet alignment torque and reluctance torque can be adjusted by changing the phase angle of the current (() as well as its magnitude. Noting that ( is measured from the q-axis in the positive (CCW) direction, we can write

and if we substitute these expressions in equation (2.60) we get

At any given current level I, we can differentiate this expression with respect to ( to find the value of ( which gives maximum torque. The result is

11_V

m is the fundamental harmonic component of the actual phase voltage. I_d ' Vq ! E X_d ' V_m cos * ! E X_d ; Iq ' !V_d X_q ' V_m sin * X_q . (2.66)

Fig. 2.33 Current-limit circle and voltage-limit circle at the change-over speed; non-salient-pole motor with X_d = X_q.

V_m2 _{' V}

d2 % Vq2, (2.67)

(X_qI_q)2 _{% (E % X}

dId)2 ' Vm2. (2.68)

If there is no saliency )Q = 0 and from eqn. (2.64) the phase angle that gives maximum torque is ( = 0: i.e., the current must be oriented in the q-axis ($ = 90E) in phase with the EMF E. Phase advance (( > 0) in a surface-magnet motor reduces the torque constant kT, which is defined as Te/I. This can be seen

in eqn. (2.64) (with L_d = L_q). The reduction in k_T is in the ratio cos (.

The argument leading to the optimum phase angle given by eqn. (2.65) assumes that the current I has a certain value. If this is the rated r.m.s. current Im, the tip of the current phasor lies on a circle, centre

0, radius I_m, Fig. 2.33. In practical terms this corresponds to the case where the drive is operating under current limit and the current regulators have complete control of the current waveform.

The EMF E is equal to TQ_1Md and at high speed it approaches the maximum available supply voltage V_m. At certain values of ( the drive may not have sufficient voltage to maintain the current Im. We must

therefore examine what happens when the drive is voltage-limited rather than current-limited. Imagine that the drive is supplying maximum voltage V_m to each phase of the motor, such that the phase angle between Vm and E is * (Fig. 2.32).11 From the phasor diagram

and so that

T_e ' mp T E V_m X_d sin * % V_m2 2 1 X_d ! 1 X_q sin 2* . (2.69) * ' cos!1 ! J ± J2 % 8 4 (2.70) (V_m/X_d)2 _{' (E /X} d)2 % Im2 (2.72) J ' E /Vm 1 ! X_d/X_q. (2.71) T_Q ' Vm Q_1Md2 _{% (L} dIm)2 rad/sec. _(2.73)

According to these equations, the tip of the voltage phasor lies on a circle of radius Vm, but the tip of

the current phasor has an elliptical locus as * and ( vary. Under this type of control, with constant (maximum) voltage V_m but variable phase angle * between V_m and E, the torque can be calculated by substituting eqns. (2.66) in eqn. (2.62) to eliminate Id and Iq. The result is

Again we can differentiate this expression to find the phase angle * which maximises the torque. After some simplification the result is

where

If there is no saliency, the angle which gives maximum torque is * = 90E. The phase angle of the current will vary in a more complex manner as * varies.

For machines with no saliency (such as surface-magnet motors), Xd = Xq and we can summarise the

guidelines for maximum torque as follows:

Low speed Control current with ( = 0 High speed Control voltage with * = 90E

TABLE 2.5

The change-over between “low speed” and “high speed” is the maximum speed at which the rated current Im can be driven into the motor with ( = 0. For a surface-magnet (non-salient-pole) motor with

X_d = X_q the voltage-limit ellipse becomes a circle, and the change-over frequency can be determined from its intersection Q with the current-limit circle, Fig. 2.33. At this speed the voltage-limit circle is just large enough to intersect the current-limit circle at point Q, where Iq = Im and Id = 0. Therefore

In this equation Xd and E are both proportional to speed or frequency T, but Im and Vm are fixed. If we

substitute E = TQ_1Md and X_d = TL_d we can re-arrange eqn. (2.72) to give the change-over frequency:

The subscript Q identifies this value of frequency as the change-over value, sometimes known as the corner-point or base value. The corresponding speed in rpm is N_Q = T_Q /p × 30/B.

At speeds higher than NQ it is still possible to drive rated current Im into the motor, but not at the

optimum angle for maximum torque (( = 0 in a non-salient-pole motor). As the frequency increases the radius of the voltage-limit circle decreases and the intersection with the rated-current circle moves along the arc QD. To illustrate this, Fig. 2.34 shows the conditions at four different speeds, N1 < N2 (=

N_Q N_D ' u ! 1 ! u 2 (2.74) 1 2 < u < 1 (2.75)

Fig. 2.34 Circle diagram at four different speeds (non-salient-pole motor)

At low speed (circle 1) the voltage-limit circle completely encloses the current-limit circle at I_m, which means that this current can be driven into the motor with any phase angle. In fact the current could be increased up to the value OL, approximately twice Im, under the control of the current regulator.

As the speed increases the voltage-limit circle shrinks, until at N_Q the maximum current that can be driven along the q-axis is Im, circle 2. At a still higher speed circle 3 shows operation at P with I = Im,

but the phase angle ( is advanced as shown, and the torque is reduced by the factor cos (. Eventually a speed is reached at which the current I_m can be driven only along the negative d axis, circle 4. All the current is now used to suppress the flux (flux weakening), and none of it is available to produce torque. The intersection is at point D and the torque is zero. To achieve this point the current regulator must operate with a massive phase advance of 90E and with maximum current reference. It can be shown that the speed at which point D is reached is related to the change-over speed by

where u = TQQ1Md/Vm = EQ/Vm, EQ being the value of E at the change-over speed. For a solution to

exist at a positive speed, we must have

For example, if u = 0.8, N_D = 5 N_Q, but if u = 0.9, N_D = 2.155 N_Q. Alternatively suppose that the motor must maintain maximum torque at speeds up to 3,000 rpm and be capable of just reaching 6,000 rpm. Then ND/NQ = 2 and according to equation 15, u must be no higher than 0.911.

The ratio of the speeds ND and NQ can be expressed in terms of the reactance or inductance of the

T_e ' mp T IdNIq)X (2.78) )X ' X_d ! X_q and I_dN ' I_d % E )X. (2.79) x ' LdIm Q_1Md. (2.76) N_D N_Q ' x2 _{% 1} x ! 1 . (2.77)

Fig. 2.35 Constant-torque loci (salient-pole motor)

The per-unit synchronous reactance is identical to x, and by equating CD to OP in Fig. 2.34 we get

Even with x = 0.3 (a high value for a surface-magnet motor), ND/NQ = 1.491, which is quite a narrow

speed range above the corner-point speed. For a motor which must maintain constant torque up to 3,000 rpm and just be able to reach 6,000 rpm ( ND/NQ = 2), eqn. (2.77) prescribes that x must be at least

2.347, which is unrealistically high. This helps to explain why in some instances, additional inductance has been connected in series with the motor to extend the speed range above the corner-point speed. The additional inductance makes the system more responsive to phase advance in weakening the total flux, but some of this flux is in the external inductance and not in the motor. Note that eqns. (2.74) and (2.77) are independent and must both be satisfied.

Although eqns. (2.74) and (2.77) are an incomplete account of the variation of torque with speed, and apply only to non-salient-pole (surface-magnet) motors, they help to resolve the question as to whether phase advance is needed “to overcome the rising EMF” or “to compensate for inductance” as the speed increases. The answer is both.

A constant-torque locus can be superimposed on the circle diagram, as in Fig. 2.35, by writing the torque equation as

where

This is a rectangular hyperbola asymptotic to the negative d-axis and to a q-axis which is shifted to the right by E/)X. With high-energy magnets the constant-torque contours are more nearly horizontal, but with low-energy magnets they have more curvature. Fig. 2.35 is drawn for a salient-pole motor (Xd

12_{See Strauss F, Synchronous machines with rotating permanent-magnet fields, AIEE Transactions, Vol. 71, Pt. II, pp. 887-893,} October 1952. Also Merrill FW, Permanent-magnet excited synchronous motors, Transactions AIEE, Vol. 74, pp. 1754 1760, 1955. 13_{See DD Hershberger, Design considerations of fractional horsepower size permanent-magnet motors and generators, AIEE} Transactions, pp. 581!584, June 1953. ( ' sin!1 1 4 ! 0.9 1.0 % 0.9 1.0 2 % 8 ' 31.13E. (2.81) I ' I_d % j I_q ' I ej$ ' I ej(( % 90E); V ' V_d % j V_q ' V ej(* % 90E)_. (2.82) ( ' sin!1 1 4 ! u )x % u )x 2 % 8 . (2.80)

The constant-torque loci in Fig. 2.35 are drawn for three torques T₁ < T₂ < T₃. The middle one goes through the point Q which we earlier associated with the corner-point for a nonsalient pole motor. With saliency, however, the constant-torque loci show that the torque can be increased by phase advance between Q and P, with the current maintained at the rated value I_m. The additional torque is reluctance torque, the second term in eqn. (2.64), which comes from the saliency. As ( increases, the sin ( term increases quickly from zero while cos ( changes only slowly. As we have seen, eqn. (2.65) can be used to determine the phase advance angle which maximises the torque. Having defined the per-unit synchronous reactance x for a non-salient-pole motor, it is a simple matter to extend this to

xd and xq, the per-unit synchronous reactances in the d- and q-axes respectively, and put )x = (xd ! xq).

If Im is the rated current, its per-unit value can be taken as 1, and if we use the per-unit EMF u as

defined earlier, eqn. (2.65) gives

For example, suppose u = 0.9 and xd = 0.5 and xq = 1.5. Then )x = !1.0 and

The phase advance can be used to achieve the same torque at a lower current, or the same torque at a higher speed as the voltage-limit ellipse shrinks in size.

So far we have discussed the controlled variation of the current or voltage in terms of their magnitudes and phase angles, I and ( or V and *; in other words, we have represented the voltage and current in polar coordinates. It is equally straightforward to describe the controlled variation of current in terms of the cartesian components Id and Iq, and the controlled variation of voltage in terms of Vd and Vq.

Mathematically these components are equivalent to Ig( and Vg*. Using complex numbers,