Speed and torque control  (Page 6/19)

Figure 10.11 Variable-speed operating regimes for a synchronous motor.

• Although during steady-state operation the speed of a synchronous motor is determined by the frequency of the drive, speed control by means of frequency control is of limited use in practice. This is due in most part to the fact that it is difficult for the rotor of a synchronous machine to track arbitrary changes in the frequency of the applied armature voltage.
• Problems with changing speed result from the fact that, in order to develop torque, the rotor of a synchronous motor must remain in synchronism with the stator flux. Control of synchronous motors can be greatly enhanced by control algorithms in which the stator flux and its relationship to the rotor flux are controlled directly.

Torque control

Direct torque control in an ac machine, which can be implemented in a number of different ways, is commonly referred to as field-oriented control or vector control. To facilitate our discussion of field-oriented control, it is helpful to return to the discussion of Section 5.6.1. Under this viewpoint, which is formalized in Appendix C, stator quantities (flux, current, voltage, etc.) are resolved into components which rotate in synchronism with the rotor. Direct-axis quantities represent those components which are aligned with the field-winding axis, and quadrature-axis components are aligned perpendicular to the field-winding axis.

Section C.2 of Appendix C derives the basic machine relations in dq0 variables for a synchronous machine consisting of a field winding and a three-phase stator winding. The transformed flux-current relationships are found to be

$\lambda =L_d{i}_d+L_{\text{af}}{i}_f$ (10.15)

${\lambda }_q=L_q{i}_q$ (10.16)

${\lambda }_f=\frac{3}{2}{L}_{\text{af}}{i}_d+L_{\text{ff}}{i}_f$ (10.17)

where the subscripts d, q, and f refer to armature direct-, quadrature-axis, and fieldwinding quantities respectively. Note that throughout this chapter we will assume balanced operating conditions, in which case zero-sequence quantities will be zero and hence will be ignored.

The corresponding transformed voltage equations are

$v_d=R_a{i}_d+\frac{{\mathrm{d\lambda }}_d}{\text{dt}}-{\omega }_{\text{me}}{\lambda }_q$ (10.18)

$v_q=R_a{i}_q+\frac{{\mathrm{d\lambda }}_q}{\text{dt}}+{\omega }_{\text{me}}{\lambda }_d$ (10.19)

$v_f=R_f{i}_f+\frac{{\mathrm{d\lambda }}_f}{\text{dt}}$ (10.20)

where ${\omega }_{\text{me}}=(\text{poles}/2){\omega }_m$ is the electrical angular velocity of the rotor.

Finally, the electromagnetic torque acting on the rotor of a synchronous motor is shown to be $T_{\text{mech}}=\frac{3}{2}\left[\frac{\text{poles}}{2}\right]({\lambda }_d{i}_q-{\lambda }_q{i}_d)$ (10.21)

Under steady-state, balanced-three-phase operating conditions, ${\omega }_{\text{me}}={\omega }_e$ where ${\omega }_e$ is the electrical frequency of the armature voltage and current in rad/sec. Because the armature-produced mmf and flux waves rotate in synchronism with the rotor and hence with respect to the dq reference frame, under these conditions an observer in the dq reference frame will see constant fluxes, and hence one can set d/dt = 0

Letting the subscripts F, D, and Q indicate the corresponding constant values of field-, direct- and quadrature-axis quantities respectively, the flux-current relationships of Eqs. 10.15 through 10.17 then become

${\lambda }_D=L_d{i}_D+L_{\text{af}}{i}_F$ (10.22)

${\lambda }_Q=L_q{i}_Q$ (10.23)

${\lambda }_F=\frac{3}{2}{L}_{\text{af}}{i}_D+L_{\text{ff}}{i}_F$ (10.24)

Armature resistance is typically quite small, and, if we neglect it, the steady-state voltage equations (Eqs. 10.18 through 10.20) then become