Quaternions and Vector Mapping in Three-Dimensional Space

2012 ◽  
Vol 235 ◽  
pp. 101-106
Author(s):  
Gui San Li
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Unit Vector ◽  
Geometric Representation ◽  
Space Vector ◽  
Vector Operation ◽  
Spatial Mechanism ◽  
Multiplication Operation ◽  
Vector Mapping ◽  
Three Dimensional Space

On the basis of analyzing the relations between vector operation and properties of multiplication operation of quaternions, the geometric representation and mapping of quaternions and vectors defined in three-dimensional space are established. The mathmatic operation of space vector, in addition to any space unit vector, rotating around three coordinate axises is carried based on the utilization of quaternions.The main purpose of this paper is to solve the issues of mathmatic tools introduced in spatial mechanism analysises and integrations by making use of quatnions.

Vectors

1996 ◽  
Author(s):  
Gerhard Oertel
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Unit Vector ◽  
Cross Product ◽  
Nonzero Vector ◽  
Algebraic Representation ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Three Dimensional Space

The reader, even if familiar with vectors, will find it useful to work through this chapter because it introduces notation that will be used throughout this book. We will take vectors to be entities that possess magnitude, orientation, and sense in three-dimensional space. Graphically, we will represent them as arrows with the sense from tail to head, magnitude proportional to the length, and orientation indicated by the angles they form with a given set of reference directions. Two different kinds of symbol will be used to designate vectors algebraically, boldface letters (and the boldface number zero for a vector of zero magnitude), and subscripted letters to be introduced later. The first problems deal with simple vector geometry and its algebraic representation. Multiplying a vector by a scalar affects only its magnitude (length) without changing its direction. Problem 1. State the necessary and sufficient conditions for the three vectors A, B, and C to form a triangle. (Problems 1–9, 12–14, 19–23, and 25 from Sokolnikoff & Redheffer, 1958.) Problem 2. Given the sum S = A + B and the difference D = A – B, find A and B in terms of S and D (a) graphically and (b) algebraically. Problem 3. (a) State the unit vector a with the same direction as a nonzero vector A. (b) Let two nonzero vectors A and B issue from the same point, forming an angle between them; using the result of (a), find a vector that bisects this angle. Problem 4. Using vector methods, show that a line from one of the vertices of a parallelogram to the midpoint of one of the nonadjacent sides trisects one of the diagonals. Two vectors are said to form with each other two distinct products: a scalar, the dot product, and a vector, the cross product.

A Fast Digital 3D-SVPWM Algorithm for Four-Leg Converter

2014 ◽  
Vol 541-542 ◽  
pp. 494-497
Author(s):  
Peng Xian Song ◽  
Ping Wang ◽  
Yao Hua Li
Keyword(s):  
Mathematical Model ◽  
Computer Simulation ◽  
Coordinate Transformation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Space Vector Pwm ◽  
Space Vector ◽  
Computational Burden ◽  
Three Phase ◽  
Three Dimensional Space

The three-phase four-leg converter can be obtained by adding a bridge into the traditional three-phase three-leg converter. The average mathematical model of three-phase four-leg converter is described in this paper. The three-dimensional space vector PWM (3D-SVPWM) is analyzed. For simplifying the modulation, a fast digital SVPWM algorithm which eliminates coordinate transformation and saves calculating time was proposed. The feasibility of the proposed modulation technique is verified by computer simulation. These results show that the proposed fast digital 3D-SVPWM technique can be easily implemented without conventional computational burden.

Three-Dimensional Space Vector Modulation Strategy for Capacitor Balancing in Split Inductor Neutral-Point Clamped Multilevel Inverters

2018 ◽  
Vol 27 (14) ◽  
pp. 1850232 ◽  
Author(s):  
Palanisamy Ramasamy ◽  
Vijayakumar Krishnasamy ◽  
Mohamed Ali Jagabar Sathik ◽  
Ziad M. Ali ◽  
Shady H. E. Abdel Aleem
Keyword(s):  
Dimensional Space ◽  
Computational Cost ◽  
Three Dimensional ◽  
Neutral Point ◽  
Space Vector Modulation ◽  
Space Vector ◽  
Field Programmable ◽  
Vector Modulation ◽  
Switching Method ◽  
Three Dimensional Space

Capacitor imbalance is one of the major drawbacks of a neutral-point clamped multilevel inverter (NPC-MLI). The capacitor imbalance rises due to the nonuniform switching, nonideal DC link capacitors, improper commutation, and various asymmetrical phase currents in switching states. The imbalance can be minimized by using proper switching using a redundancy switching method and avoiding the usage of medium vectors. The computational cost of this system is to be decided by the level of the inverter system. This paper presents a comprehensive three-dimensional space vector modulation (3D-SVM) strategy to eliminate the imbalance in the DC link capacitor voltage, which is across the applied input DC source. The technique is easy to implement without using any trigonometric functions, lookup tables, or angle determinations. This proposed scheme has been verified using MATLAB Simulink and authenticated using field-programmable gate array (FPGA) controller.

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