Scalar Induction Machine Control

11.2 Scalar Induction Machine Control

Induction machine scalar control is derived using the induction machine steady-state model shown in Fig. 11.1 [1]. The phasor form of the machine voltages and currents is indicated by capital letters. The

stator series resistance and leakage reactance are R 1 and X 1 , respectively. The referred rotor series resistance and leakage reactance are R 2 and X 2 , respectively. The magnetizing reactance is X m ; the core loss due to eddy currents and the hysteresis of the iron core is represented by the shunt resistance R c . The machine slip s is defined as [1]

w – w = r ----------------- s e

where ω e is the synchronous, or excitation frequency, and ω r is the machine shaft speed, both in electrical radians-per-second. The power supplied to the machine shaft can be expressed as

shaft = ---------- R 1 2 i 2 (11.2)

Solving for I 2 and using Eq. (11.2), the shaft torque can be expressed as

T e = ------------------------------------------------------------------------- 3V in R 2 2 s 2 2 (11.3)

w e [ ( sR 1 + R 2 ) + s ( X 1 + X 2 ) ]

where the numeral 3 in the numerator is used to include the torque from all three phases. This expression makes clear that induction machine torque control is possible by varying the magnitude of the applied stator voltage. The normalized torque vs. slip curves for a typical induction machine corresponding to various stator voltage magnitudes are shown in Fig. 11.2 . Speed control is accomplished by adjusting the input voltage until the machine torque for a given slip matches the load torque. However, the developed torque decreases as the square of the input voltage, but the rotor current decreases linearly with the input voltage. This operation is inefficient and requires that the load torque decrease with decreasing machine speed to prevent overheating [1, 2]. In addition, the breakdown torque of the machine decreases as the square of the input voltage. Fans and pumps are appropriate loads for this type of speed control because the torque required to drive them varies linearly or quadratically with their speed.

Linearization of Eq. (11.3) with respect to machine slip yields

2 3V 2

in s

3V in ( w e – w )

T e = -----------------

= r --------------------------------------

The characteristic torque curve can be shifted along the speed axis by changing ω e with the capability

) .m

(N

orque

V in =0.25V 0

ed t iz al

orm

V in =0.75V 0

V in =V 0

Slip

FIGURE 11.2 Normalized torque–slip curves with varying input voltage magnitudes for a typical induction machine.

for developing rated torque throughout the entire speed range given a constant stator voltage magnitude. An inverter is needed to drive the induction machine to implement frequency control.

One remaining complication is the fact that the magnetizing reactance changes linearly with excitation frequency. Therefore, with constant input voltage, the input current increases as the input frequency decreases. In addition, the stator flux magnitude increases as well, possibly saturating the machine. To prevent this from happening, the input voltage must be varied in proportion to the excitation frequency.

From Eq. (11.4), if the input voltage and frequency are proportional with proportionality constant k f , the electrical torque developed by the machine can be expressed as

3k f

------- w ( e – w r )

and demonstrates that the torque response of the machine is uniform throughout the full speed range. The block diagram for the scalar-controlled induction drive is shown in Fig. 11.3 . The inverter DC- link voltage is obtained through rectification of the AC line voltage. The drive uses a simple pulse-width- modulated (PWM) inverter whose time-average output voltages follow a reference-balanced three-phase set, the frequency and amplitude of which are provided by the speed controller. The drive shown here uses an active speed controller based on a proportional integral derivative (PID), or other type of controller. The input to the speed controller is the error between a user-specified reference speed and the shaft speed of the machine. An encoder or other speed-sensing device is required to ascertain the shaft speed. The drive can be operated in the open-loop configuration as well; however, the speed accuracy will be reduced significantly.

T Voltages

AC Line

v + dc Rectifier Induction Inverter

k f Controller ω Sense

ω ref

FIGURE 11.3 Block diagram of scalar induction machine drive. Practical scalar-controlled drives have additional functionality, some of which is added for the conve-

nience of the user. In a practical drive, the relationship between the input voltage magnitude and frequency takes the form

(11.6) where V offset is a constant. The purpose of this offset voltage is to overcome the voltage drop created by

V in = k f w e + V offset

the stator series resistance. The relationship (11.6) is usually a piecewise linear function with several breakpoints in a standard scalar-controlled drive. This allows the user to tailor the drive response characteristic to a given application.