Complex Space Vector Research Articles

A Complex Space Vector Approach to Rotor Temperature Estimation for Line-Connected Induction Machines With Impaired Cooling

Thermal models used in overload protection relays often fail to predict the rotor temperature for motors with impaired cooling conditions. In this case, the temperature estimated from the rotor resistance can be used as an indicator of the motor's cooling capability. However, this estimated rotor temperature is often corrupted by estimation error due to the asymmetry in the power supply. A detailed analysis of the trajectories of the complex current and voltage space vectors in a synchronous reference frame is presented in this paper, and the negative sequence fundamental frequency components are identified as the major cause of the estimation error. Based on this analysis, a fast and efficient algorithm is proposed to calculate the rotor temperature for line-connected induction machines. By applying the Goertzel algorithm to the complex current and voltage space vectors, constructed directly from the motor terminal measurements, the positive sequence fundamental frequency components are extracted, and the rotor temperature is then estimated with significantly reduced estimation error. Compared to the conventional scheme based on the fast Fourier transform, the proposed algorithm is faster and more efficient, and is therefore more suitable for implementation on a low-cost hardware platform.

Sensorless Control of Induction Machines—With or Without Signal Injection?

Controlled induction motor drives without mechanical speed sensors at the motor shaft have the attractions of low cost and high reliability. To replace the sensor, information on the rotor speed is extracted from measured stator currents and from voltages at motor terminals. Vector-controlled drives require estimating the magnitude and spatial orientation of the fundamental magnetic flux waves in the stator or in the rotor. Open-loop estimators or closed-loop observers are used for this purpose. They differ with respect to accuracy, robustness, and sensitivity against model parameter variations. Dynamic performance and steady-state speed accuracy around zero speed range are achieved by signal injection, exploiting the anisotropic properties of the machine. The overview in this paper uses signal flow graphs of complex space vector quantities to provide an insightful description of the systems used in sensorless control of induction motors.

Approximation algorithms for M ax-3-C ut and other problems via complex semidefinite programming

A number of recent papers on approximation algorithms have used the square roots of unity, −1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit vectors in real space using semidefinite programming in order to obtain near optimum solutions to these problems. In this paper, we consider using the cube roots of unity, 1, e i2 π/3 , and e i4 π/3 , to represent ternary decision variables for problems in combinatorial optimization. Here the natural relaxation is that of unit vectors in complex space. We use an extension of semidefinite programming to complex space to solve the natural relaxation, and use a natural extension of the random hyperplane technique introduced by the authors in Goemans and Williamson (J. ACM 42 (1995) 1115–1145) to obtain near-optimum solutions to the problems. In particular, we consider the problem of maximizing the total weight of satisfied equations x u−x v≡c ( mod 3) and inequations x u−x v≢c ( mod 3) , where x u ∈{0,1,2} for all u. This problem can be used to model the M ax-3-C ut problem and a directed variant we call M ax-3-D icut. For the general problem, we obtain a 0.793733-approximation algorithm. If the instance contains only inequations (as it does for M ax-3-C ut), we obtain a performance guarantee of 7 12 + 3 4π 2 arccos 2(−1/4)−ε>0.836008 . This compares with proven performance guarantees of 0.800217 for M ax-3-C ut (by Frieze and Jerrum (Algorithmica 18 (1997) 67–81) and 1 3 +10 −8 for the general problem (by Andersson et al. (J. Algorithms 39 (2001) 162–204)). It matches the guarantee of 0.836008 for M ax-3-C ut found independently by de Klerk et al. (On approximate graph colouring and Max- k-Cut algorithms based on the ϑ-function, Manuscript, October 2000). We show that all these algorithms are in fact equivalent in the case of M ax-3-C ut, and that our algorithm is the same as that of Andersson et al. in the more general case.

Sensorless control of induction motor drives

Controlled induction motor drives without mechanical speed sensors at the motor shaft have the attractions of low cost and high reliability. To replace the sensor the information on the rotor speed is extracted from measured stator voltages and currents at the motor terminals. Vector-controlled drives require estimating the magnitude and spatial orientation of the fundamental magnetic flux waves in the stator or in the rotor. Open-loop estimators or closed-loop observers are used for this purpose. They differ with respect to accuracy, robustness, and sensitivity against model parameter variations. Dynamic performance and steady-state speed accuracy in the low-speed range can be achieved by exploiting parasitic effects of the machine. The overview in this paper uses signal flow graphs of complex space vector quantities to provide an insightful description of the systems used in sensorless control of induction motors.

The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms

In this paper we find new surfaces with parallel mean curvature vectors in two-dimensional complex space forms that are not of constant Kähler angles and then classify all of them. The surface can be isometrically deformed preserving both the length of the mean curvature vector and the Kähler angle. From this classification, the deformations appear only when the curvature of the ambiant space is nonpositive.

State of the art of controlled AC drives without speed sensors

The operation of speed-controlled AC drives without mechanical speed or position sensors requires the estimation of internal state variables of the machine. This assessment is based exclusively on measured terminal voltages and currents. Low cost, medium performance sensorless drives can be designed using simple algebraic speed estimators. High-performance systems rely on dynamic models for the estimation of the magnitude and spatial orientation of magnetic flux waves in the stator or in the rotor. Open loop estimators and closed loop observers differ with respect to accuracy, robustness, and limits of applicability. The overview in this paper uses signal flow graphs of complex space vector quantities to give a description of the physical and mathematical systems used in sensorless control.

Time‐frequency representations of instantaneous complex power, voltage and current space vectors

AbstractThis paper introduces a new time‐frequency representation and analyses the time‐frequency structure of the instantaneous complex power, voltage and current space vectors, which are powerful mathematical tools to describe the behaviour of three‐phase systems under unbalanced and non‐sinusoidal conditions. Computer simulations are carried out so that the performance of the new time‐frequency representation can be verified.

Subspace rotation using modified Householder transforms and projection matrices — Robustness of DOA algorithms

Abstract A new Householder transformation for vectors in complex space is suggested. The transformation does not require the inner product of mirror image vectors to be real. A method, based on this transform, is designed to solve the subspace rotation problem. The method produces a unitary rotation operator. Also, another method that uses projection matrices and yields a nonunitary rotation operator is developed. Robustness of directions-of-arrival (DOA) algorithms that use focussing matrices is discussed.