An improved method and prior methods of deadbeat direct torque control involve the use of pulse-width modulation (PWM) of applied voltages. The prior methods are based on the use of stator flux and stator current as state variables, leading to mathematical solutions of control equations in forms that do not lend themselves to clear visualization of solution spaces. In contrast, the use of rotor and stator fluxes as the state variables in the present improved method lends itself to graphical representations that aid in understanding possible solutions under various operating conditions. In addition, the present improved method incorporates the superposition of high-frequency carrier signals for use in a motor-self-sensing technique for estimating the rotor shaft angle at any speed (including low or even zero speed) without need for additional shaft-angle-measuring sensors.

Prerequisite to a description of the method is a description of the concept of *dq* variables and *dq* coordinates. The "*d*" and "*q*" signify "direct" and "quadrature," respectively. The *dq* coordinates lie along two orthogonal axes attached to the stator. The *dq* coordinates and the *dq* variables (which can be, for example, phase voltages and fluxes projected onto the *dq* coordinates) are commonly used in the design and analysis of induction motors.

The derivation of the method begins with the observation that a desired change in the torque of an induction motor can be represented as a straight line with units of flux on both axes of the *d-q* plane. The equation that describes the straight line can be derived from the discrete form of the induction-motor equations by use of the stator and rotor fluxes as state variables. The desired change in stator flux can be represented as a circle on the *d-q* plane. The voltage needed to achieve the desired change in torque and change in stator flux in one time step can be found from an intersection of the torque line and stator-flux circle (see figure).

Going beyond the aforementioned graphical representations, the maximum-current operating limits can be represented as a set of two lines parallel to the torque line. The maximum-voltage operating limits can be represented as a hexagon on the *d-q* plane. The maximum-voltage hexagon can be divided into four regions corresponding to an increase or decrease in torque and an increase or decrease in flux. As a result, the effect of a particular voltage vector (increase or decrease in torque or flux) can be clearly seen.

A control algorithm based on the graphical constructions and the underlying equations has been developed. The algorithm causes the synthesis of the needed excitation voltages by use of space vector modulation techniques to calculate and command inverter duty cycles. In the implementation of the algorithm without shaft-angle-measuring sensors, the high-frequency voltage needed for sensorless operation is added to the fundamental voltage and used as input for the PWM calculations. The PWM portion of the control algorithm then determines the duty cycles to generate both voltages at once. The signal from the resulting high-frequency current is processed to estimate the angular position and speed of the rotor.

*This work was done by Barbara H. Kenny of Glenn Research Center and Robert D. Lorenz of the University of Wisconsin. *

*Inquiries concerning rights for the commercial use of this invention should be addressed to*

###### NASA Glenn Research Center

Commercial Technology Office

Attn: Steve Fedor

Mail Stop 4–8

21000 Brookpark Road

Cleveland

Ohio 44135.

*Refer to LEW-17329.*

##### Motion Control Tech Briefs Magazine

This article first appeared in the April, 2003 issue of *Motion Control Tech Briefs* Magazine.

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