This document summarizes space vector transformations of three-phase systems. It discusses symmetrical components analysis in the frequency and time domains, which allows decomposition of an unbalanced three-phase system into positive, negative, and zero sequence components. It also describes Clarke's transformation to a stationary αβ0 reference frame, which defines three independent real components from the complex positive and negative sequence components.
This document describes a simulation project of a space vector PWM inverter. It provides details of the system configuration including IGBT switches, DC link voltage, frequencies, and load components. It then provides an in-depth explanation of space vector PWM technique, including the principle of PWM, representation of voltage vectors in the dq reference frame, and algorithm for determining switching times. State-space equations for the L-C output filter are also derived. The overall purpose is to simulate and analyze a three-phase PWM inverter using space vector modulation in MATLAB/Simulink.
A three-phase system uses three sinusoidal voltages that are 120 degrees out of phase to provide constant power output. A balanced three-phase system can be analyzed using a single-phase equivalent circuit. Power in a three-phase system is measured using three wattmeters connected in a Y configuration for a Y-connected load or two wattmeters connected between line voltages for a three-wire system. The total power is the sum of the wattmeter readings.
This document summarizes key concepts about three-phase systems. It defines a three-phase system as having three sinusoidal voltages differing in phase by 120 degrees. The voltages can form a positive or negative sequence. Three-phase systems are commonly used for power generation, transmission, and distribution due to their ability to transmit more power with less material. Formulas are provided for calculating line voltages, currents, and power in balanced and unbalanced three-phase systems. Advantages of three-phase systems like constant torque and easier starting of motors are also discussed.
- The document is an exam paper for the subject Power System I. It contains two sections - Section A and Section B, with 4 questions each. Students have to answer any 3 questions from each section.
- The questions cover various topics related to power systems including transformer and transmission line modeling, fault analysis, symmetrical components, load flow analysis, relay protection and power flow through transmission lines.
- The questions involve derivations, calculations, explanations and numerical problems related to the power system topics. Diagrams and data are provided with some questions.
Distance Algorithm for Transmission Line with Mid-Point Connected STATCOMIRJET Journal
This document presents an adaptive zone selection algorithm for distance protection of a transmission line with a midpoint connected STATCOM device. The algorithm aims to address challenges to distance protection posed by the presence of the STATCOM. It investigates the impact of the STATCOM on apparent impedance seen by distance relays under different fault conditions using EMTDC/PSCAD software. An adaptive setting is proposed that calculates a new reach based on system and STATCOM parameters to ensure proper operation of distance relays for both underreach and overreach scenarios. The performance of the proposed adaptive algorithm is evaluated through simulations of various single line to ground and three phase fault cases with different fault locations, resistances and system load angles.
Application of SVM Technique for Three Phase Three Leg Ac/Ac Converter TopologyIOSR Journals
This paper presents a simulation of a three-phase three-leg AC/AC converter topology using nine IGBTs and space vector pulse width modulation (SVM) technique. The proposed topology reduces the number of switches compared to conventional back-to-back and matrix converters. Simulation results show the converter provides sinusoidal input and output voltages with unity power factor under constant frequency and variable frequency operation. Experimental results from a 5kVA prototype verify the validity of the proposed scheme.
This document summarizes Nathan Wendt's final project for EE321, which involved designing third-order passive frequency-selective circuits. Section I derives the general transfer function and analyzes low-pass behavior. Section II examines the low-pass frequency response and Butterworth design. Section III designs a high-pass Butterworth filter. MATLAB is used throughout to simulate and analyze the circuit designs.
This document describes a simulation project of a space vector PWM inverter. It provides details of the system configuration including IGBT switches, DC link voltage, frequencies, and load components. It then provides an in-depth explanation of space vector PWM technique, including the principle of PWM, representation of voltage vectors in the dq reference frame, and algorithm for determining switching times. State-space equations for the L-C output filter are also derived. The overall purpose is to simulate and analyze a three-phase PWM inverter using space vector modulation in MATLAB/Simulink.
A three-phase system uses three sinusoidal voltages that are 120 degrees out of phase to provide constant power output. A balanced three-phase system can be analyzed using a single-phase equivalent circuit. Power in a three-phase system is measured using three wattmeters connected in a Y configuration for a Y-connected load or two wattmeters connected between line voltages for a three-wire system. The total power is the sum of the wattmeter readings.
This document summarizes key concepts about three-phase systems. It defines a three-phase system as having three sinusoidal voltages differing in phase by 120 degrees. The voltages can form a positive or negative sequence. Three-phase systems are commonly used for power generation, transmission, and distribution due to their ability to transmit more power with less material. Formulas are provided for calculating line voltages, currents, and power in balanced and unbalanced three-phase systems. Advantages of three-phase systems like constant torque and easier starting of motors are also discussed.
- The document is an exam paper for the subject Power System I. It contains two sections - Section A and Section B, with 4 questions each. Students have to answer any 3 questions from each section.
- The questions cover various topics related to power systems including transformer and transmission line modeling, fault analysis, symmetrical components, load flow analysis, relay protection and power flow through transmission lines.
- The questions involve derivations, calculations, explanations and numerical problems related to the power system topics. Diagrams and data are provided with some questions.
Distance Algorithm for Transmission Line with Mid-Point Connected STATCOMIRJET Journal
This document presents an adaptive zone selection algorithm for distance protection of a transmission line with a midpoint connected STATCOM device. The algorithm aims to address challenges to distance protection posed by the presence of the STATCOM. It investigates the impact of the STATCOM on apparent impedance seen by distance relays under different fault conditions using EMTDC/PSCAD software. An adaptive setting is proposed that calculates a new reach based on system and STATCOM parameters to ensure proper operation of distance relays for both underreach and overreach scenarios. The performance of the proposed adaptive algorithm is evaluated through simulations of various single line to ground and three phase fault cases with different fault locations, resistances and system load angles.
Application of SVM Technique for Three Phase Three Leg Ac/Ac Converter TopologyIOSR Journals
This paper presents a simulation of a three-phase three-leg AC/AC converter topology using nine IGBTs and space vector pulse width modulation (SVM) technique. The proposed topology reduces the number of switches compared to conventional back-to-back and matrix converters. Simulation results show the converter provides sinusoidal input and output voltages with unity power factor under constant frequency and variable frequency operation. Experimental results from a 5kVA prototype verify the validity of the proposed scheme.
This document summarizes Nathan Wendt's final project for EE321, which involved designing third-order passive frequency-selective circuits. Section I derives the general transfer function and analyzes low-pass behavior. Section II examines the low-pass frequency response and Butterworth design. Section III designs a high-pass Butterworth filter. MATLAB is used throughout to simulate and analyze the circuit designs.
This document discusses balanced three-phase delta-connected loads. It covers calculating voltages, currents, and power in delta-connected circuits. The key learning goals are understanding basic delta connections, calculating voltages and currents in balanced delta loads, and calculating complex power. Examples are provided to demonstrate calculating phase and line currents and drawing phasor diagrams for delta loads.
This document contains an exam paper for the subject Power System I (EEE 305). It has two sections with multiple choice and descriptive questions. Section A contains questions related to power system modeling, symmetrical components, transmission lines and fault analysis. Section B contains questions on load flow analysis, transmission lines, and power flow control. The paper tests students' understanding of fundamental power system concepts and their ability to analyze common power system problems.
Ekeeda Provides Online Electrical and Electronics Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
This document provides an overview of complex power in electrical systems. It defines phasor representation using complex exponentials to simplify analysis of constant frequency AC circuits. It describes how real and reactive power can be calculated from voltage and current phasors and discusses power factor. The document also discusses reactive compensation using capacitors to improve power factor by supplying reactive power locally. It provides an example of power factor correction and introduces balanced three-phase power systems with both wye and delta connections.
This document provides an introduction to analyzing DC resistive circuits that contain nonlinear elements using load-line analysis. It defines linear and nonlinear voltage-current characteristics and explains how load-line analysis can be used to determine the operating point of a circuit when a nonlinear element is present. Load-line analysis graphs the characteristic curves of the nonlinear element and equivalent circuit to find their intersection point, which represents the operating point. The document provides examples of using both graphical load-line analysis and analytical methods to solve circuits with nonlinear elements like diodes and thermistors.
The document discusses analyzing a single-phase power system and its theoretical variations through per unit analysis using MATLAB. It provides the theory behind per unit analysis and calculates the per unit values of the system parameters. It then manually solves the system using per unit analysis and compares the results to those obtained through simulation in MATLAB.
This document contains 20 multiple choice questions from GATE EE 2007 exam. It provides the questions, solutions and explanations for each question in detail. The key details are:
- The document contains one mark multiple choice questions on topics like transistors, voltage regulators, transformer connections, generator scheduling, HVDC systems, electric field distribution, op-amps, system stability, time variant systems, sampling, matrix rank, thyristor converters etc.
- For each question, the solutions clearly explain the concepts and calculations involved in arriving at the correct option. Relevant circuit diagrams and equations are included.
- The level of detail in the solutions and explanations is quite high as it is aimed to help students prepare for competitive exams
Ekeeda Provides Online Electronics Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree. Visit us: https://ekeeda.com/streamdetails/stream/Electronics-Engineering
This document provides a critical analysis of capacitor-commutated converters (CCC). CCCs are line-commutated three-phase bridge converters that have capacitors in series with the AC phase connections. This reduces reactive power requirements and allows commutation at firing angles normally not possible.
The document presents the operating characteristics of CCCs, including valve voltage stress and attainable control range over different load currents. Mathematical formulations are provided for the DC voltage-current characteristic and commutation angles as a function of current. Equations are derived describing the commutation process and conditions required for commutation to occur.
The document provides information about a course on power systems analysis and protection. It includes:
1. An overview of topics covered in the course including per-unit systems, power flow analysis, fault analysis, stability, and protection schemes.
2. Expected learning outcomes including analyzing balanced and unbalanced faults, demonstrating power flow software, and expressing suitable protection schemes.
3. A lecture plan outlining the contents to be covered each week.
4. Assessment details including oral exams, written tests, assignments, and a final exam.
ECNG 3015 Industrial and Commercial Electrical SystemsChandrabhan Sharma
This document discusses symmetrical components and symmetrical component networks which are used to analyze unbalanced faults in power systems. It explains that a 3-phase unbalanced system can be represented as three balanced systems known as positive, negative, and zero sequence networks. It provides details on constructing these networks for different system components like generators, transmission lines, and transformers. The networks are then used to calculate fault currents and voltages under different fault conditions.
This document discusses reactive power and voltage control. It covers topics such as the generation and absorption of reactive power, excitation systems, static and dynamic analysis, stability compensation, and various methods of voltage control including tap-changing transformers, static VAR compensators, and FACTS devices. Excitation systems are modeled and the closed-loop automatic voltage regulator model is derived. Static and dynamic analyses of the voltage regulator loop are presented to evaluate stability and response.
The document describes the design and simulation of three-phase three-level diode clamped and improved inverter configurations feeding an asynchronous motor drive. It discusses the operation of different multilevel inverter topologies including diode clamped and improved inverters. The topologies are analyzed under single phase one leg voltage circuit analysis. A split-single phase asynchronous motor model is used as the load. Simulation results of the inverter-motor drive systems using Matlab/Simulink are presented, including rotor currents, stator currents, rotor speed, electromagnetic torque, and three-phase output voltages. The number of circuit components in each inverter topology is also compared.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
This document presents an implementation of space vector modulation (SVM) for a two-level three-phase inverter using a dSPACE DS1104 controller. It describes the principles of SVM, including voltage vector modeling, sector detection, and pulse generation. Hardware experiments were conducted to validate a SVM control algorithm developed in Simulink. Results showed line voltages from the real hardware matched simulation. THD comparisons confirmed SVM provides lower distortion and higher fundamental output than sinusoidal PWM. The dSPACE system allows real-time testing of control algorithms on actual hardware.
Power Circuits and Transforers-Unit 6 Labvolt Student Manualphase3-120A
This document provides instruction on analyzing balanced three-phase AC circuits connected in wye and delta configurations. It discusses the differences between line and phase voltages and currents. Formulas are presented for calculating active, reactive, and apparent power in balanced three-phase circuits. Exercises are included to measure voltages and currents in wye- and delta-connected resistive loads to verify the theoretical calculations and relationships between line and phase values.
This document presents a design of power system stabilizer (PSS) and static var compensator (SVC) controllers using particle swarm optimization (PSO) algorithm to improve power system stability. The controllers are tested on a single machine infinite bus system model in MATLAB/Simulink. Simulation results show that the proposed PSS and SVC controller designed using PSO effectively damps out oscillations under different disturbances like balanced/unbalanced faults and small disturbances, improving stability performance compared to conventional PSS alone. The parameters of the controllers are optimized using PSO to maximize efficiency. Nonlinear time-domain simulations demonstrate the effectiveness and robustness of the proposed control scheme in enhancing power system stability over a wide range of operating conditions.
This document discusses issues related to integrating distributed energy resources (DER) into electric power grids. It provides background on DER definitions and classifications. It addresses grid integration and interconnection standards, including requirements for steady state and transient operations, protection, and islanding detection. The document outlines considerations for DER protection and control requirements to safely interconnect DER while maintaining grid reliability. It also presents examples of DER installations and research on advanced relay techniques for islanding detection.
This document discusses unintentional islanding of power systems with distributed resources like solar PV. It defines intentional and unintentional islands, and issues with unintentional islands like safety hazards, overvoltages, and loss of protection. Methods to detect unintentional islands are described, like reverse power relays and active techniques. Simulation results show one technique detecting an island within 0.5 cycles. Guidelines for assessing islanding risk are provided, and the future of anti-islanding techniques discussed, like the potential need for multiple active methods with reduced grid stiffness.
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Similar to Grid Converters for Photovoltaic and Wind Power Systems - 2010 - Teodorescu - Appendix A Space Vector Transformations of.pdf
This document discusses balanced three-phase delta-connected loads. It covers calculating voltages, currents, and power in delta-connected circuits. The key learning goals are understanding basic delta connections, calculating voltages and currents in balanced delta loads, and calculating complex power. Examples are provided to demonstrate calculating phase and line currents and drawing phasor diagrams for delta loads.
This document contains an exam paper for the subject Power System I (EEE 305). It has two sections with multiple choice and descriptive questions. Section A contains questions related to power system modeling, symmetrical components, transmission lines and fault analysis. Section B contains questions on load flow analysis, transmission lines, and power flow control. The paper tests students' understanding of fundamental power system concepts and their ability to analyze common power system problems.
Ekeeda Provides Online Electrical and Electronics Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
This document provides an overview of complex power in electrical systems. It defines phasor representation using complex exponentials to simplify analysis of constant frequency AC circuits. It describes how real and reactive power can be calculated from voltage and current phasors and discusses power factor. The document also discusses reactive compensation using capacitors to improve power factor by supplying reactive power locally. It provides an example of power factor correction and introduces balanced three-phase power systems with both wye and delta connections.
This document provides an introduction to analyzing DC resistive circuits that contain nonlinear elements using load-line analysis. It defines linear and nonlinear voltage-current characteristics and explains how load-line analysis can be used to determine the operating point of a circuit when a nonlinear element is present. Load-line analysis graphs the characteristic curves of the nonlinear element and equivalent circuit to find their intersection point, which represents the operating point. The document provides examples of using both graphical load-line analysis and analytical methods to solve circuits with nonlinear elements like diodes and thermistors.
The document discusses analyzing a single-phase power system and its theoretical variations through per unit analysis using MATLAB. It provides the theory behind per unit analysis and calculates the per unit values of the system parameters. It then manually solves the system using per unit analysis and compares the results to those obtained through simulation in MATLAB.
This document contains 20 multiple choice questions from GATE EE 2007 exam. It provides the questions, solutions and explanations for each question in detail. The key details are:
- The document contains one mark multiple choice questions on topics like transistors, voltage regulators, transformer connections, generator scheduling, HVDC systems, electric field distribution, op-amps, system stability, time variant systems, sampling, matrix rank, thyristor converters etc.
- For each question, the solutions clearly explain the concepts and calculations involved in arriving at the correct option. Relevant circuit diagrams and equations are included.
- The level of detail in the solutions and explanations is quite high as it is aimed to help students prepare for competitive exams
Ekeeda Provides Online Electronics Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree. Visit us: https://ekeeda.com/streamdetails/stream/Electronics-Engineering
This document provides a critical analysis of capacitor-commutated converters (CCC). CCCs are line-commutated three-phase bridge converters that have capacitors in series with the AC phase connections. This reduces reactive power requirements and allows commutation at firing angles normally not possible.
The document presents the operating characteristics of CCCs, including valve voltage stress and attainable control range over different load currents. Mathematical formulations are provided for the DC voltage-current characteristic and commutation angles as a function of current. Equations are derived describing the commutation process and conditions required for commutation to occur.
The document provides information about a course on power systems analysis and protection. It includes:
1. An overview of topics covered in the course including per-unit systems, power flow analysis, fault analysis, stability, and protection schemes.
2. Expected learning outcomes including analyzing balanced and unbalanced faults, demonstrating power flow software, and expressing suitable protection schemes.
3. A lecture plan outlining the contents to be covered each week.
4. Assessment details including oral exams, written tests, assignments, and a final exam.
ECNG 3015 Industrial and Commercial Electrical SystemsChandrabhan Sharma
This document discusses symmetrical components and symmetrical component networks which are used to analyze unbalanced faults in power systems. It explains that a 3-phase unbalanced system can be represented as three balanced systems known as positive, negative, and zero sequence networks. It provides details on constructing these networks for different system components like generators, transmission lines, and transformers. The networks are then used to calculate fault currents and voltages under different fault conditions.
This document discusses reactive power and voltage control. It covers topics such as the generation and absorption of reactive power, excitation systems, static and dynamic analysis, stability compensation, and various methods of voltage control including tap-changing transformers, static VAR compensators, and FACTS devices. Excitation systems are modeled and the closed-loop automatic voltage regulator model is derived. Static and dynamic analyses of the voltage regulator loop are presented to evaluate stability and response.
The document describes the design and simulation of three-phase three-level diode clamped and improved inverter configurations feeding an asynchronous motor drive. It discusses the operation of different multilevel inverter topologies including diode clamped and improved inverters. The topologies are analyzed under single phase one leg voltage circuit analysis. A split-single phase asynchronous motor model is used as the load. Simulation results of the inverter-motor drive systems using Matlab/Simulink are presented, including rotor currents, stator currents, rotor speed, electromagnetic torque, and three-phase output voltages. The number of circuit components in each inverter topology is also compared.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
This document presents an implementation of space vector modulation (SVM) for a two-level three-phase inverter using a dSPACE DS1104 controller. It describes the principles of SVM, including voltage vector modeling, sector detection, and pulse generation. Hardware experiments were conducted to validate a SVM control algorithm developed in Simulink. Results showed line voltages from the real hardware matched simulation. THD comparisons confirmed SVM provides lower distortion and higher fundamental output than sinusoidal PWM. The dSPACE system allows real-time testing of control algorithms on actual hardware.
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This document provides instruction on analyzing balanced three-phase AC circuits connected in wye and delta configurations. It discusses the differences between line and phase voltages and currents. Formulas are presented for calculating active, reactive, and apparent power in balanced three-phase circuits. Exercises are included to measure voltages and currents in wye- and delta-connected resistive loads to verify the theoretical calculations and relationships between line and phase values.
This document presents a design of power system stabilizer (PSS) and static var compensator (SVC) controllers using particle swarm optimization (PSO) algorithm to improve power system stability. The controllers are tested on a single machine infinite bus system model in MATLAB/Simulink. Simulation results show that the proposed PSS and SVC controller designed using PSO effectively damps out oscillations under different disturbances like balanced/unbalanced faults and small disturbances, improving stability performance compared to conventional PSS alone. The parameters of the controllers are optimized using PSO to maximize efficiency. Nonlinear time-domain simulations demonstrate the effectiveness and robustness of the proposed control scheme in enhancing power system stability over a wide range of operating conditions.
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This document discusses issues related to integrating distributed energy resources (DER) into electric power grids. It provides background on DER definitions and classifications. It addresses grid integration and interconnection standards, including requirements for steady state and transient operations, protection, and islanding detection. The document outlines considerations for DER protection and control requirements to safely interconnect DER while maintaining grid reliability. It also presents examples of DER installations and research on advanced relay techniques for islanding detection.
This document discusses unintentional islanding of power systems with distributed resources like solar PV. It defines intentional and unintentional islands, and issues with unintentional islands like safety hazards, overvoltages, and loss of protection. Methods to detect unintentional islands are described, like reverse power relays and active techniques. Simulation results show one technique detecting an island within 0.5 cycles. Guidelines for assessing islanding risk are provided, and the future of anti-islanding techniques discussed, like the potential need for multiple active methods with reduced grid stiffness.
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This document describes research into using a fractional order PID (FOPID) controller to control voltage fluctuations in an islanded microgrid with a single power source. The proposed FOPID controller has more tuning parameters than a standard PID controller, allowing it to better regulate the microgrid's output voltage under different load conditions and uncertainties. The controller is designed using an optimization technique to maximize system performance. Simulation results show the FOPID controller is effective at reducing voltage fluctuations and provides a fast, robust response for the microgrid system.
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This document discusses space vector pulse width modulation (SVPWM) techniques for multilevel inverters. It begins by introducing SVPWM and explaining that it provides advantages over sinusoidal PWM, including better fundamental output voltage and improved harmonic performance. It then defines two-dimensional and three-dimensional space vectors and discusses how SVPWM can be implemented in both 2D and 3D. The document focuses on SVPWM for three-leg voltage source inverters, describing the voltage space vectors and how SVPWM can be used to synthesize the required output voltage vector through PWM of the switching state vectors.
This document provides an introduction to using the ATP/EMTP software for simulating electrical systems transients. It discusses how to create data files for simulations using either ATPDraw, a graphical preprocessor, or by directly editing text files. It also describes running simulations and viewing results. The document is intended to introduce beginners to the ATP/EMTP software by following existing manuals on its usage and capabilities. It emphasizes that expertise requires experience working with the software and suggests consulting manuals and experts for in-depth knowledge of complex simulations.
TIME DIVISION MULTIPLEXING TECHNIQUE FOR COMMUNICATION SYSTEMHODECEDSIET
Time Division Multiplexing (TDM) is a method of transmitting multiple signals over a single communication channel by dividing the signal into many segments, each having a very short duration of time. These time slots are then allocated to different data streams, allowing multiple signals to share the same transmission medium efficiently. TDM is widely used in telecommunications and data communication systems.
### How TDM Works
1. **Time Slots Allocation**: The core principle of TDM is to assign distinct time slots to each signal. During each time slot, the respective signal is transmitted, and then the process repeats cyclically. For example, if there are four signals to be transmitted, the TDM cycle will divide time into four slots, each assigned to one signal.
2. **Synchronization**: Synchronization is crucial in TDM systems to ensure that the signals are correctly aligned with their respective time slots. Both the transmitter and receiver must be synchronized to avoid any overlap or loss of data. This synchronization is typically maintained by a clock signal that ensures time slots are accurately aligned.
3. **Frame Structure**: TDM data is organized into frames, where each frame consists of a set of time slots. Each frame is repeated at regular intervals, ensuring continuous transmission of data streams. The frame structure helps in managing the data streams and maintaining the synchronization between the transmitter and receiver.
4. **Multiplexer and Demultiplexer**: At the transmitting end, a multiplexer combines multiple input signals into a single composite signal by assigning each signal to a specific time slot. At the receiving end, a demultiplexer separates the composite signal back into individual signals based on their respective time slots.
### Types of TDM
1. **Synchronous TDM**: In synchronous TDM, time slots are pre-assigned to each signal, regardless of whether the signal has data to transmit or not. This can lead to inefficiencies if some time slots remain empty due to the absence of data.
2. **Asynchronous TDM (or Statistical TDM)**: Asynchronous TDM addresses the inefficiencies of synchronous TDM by allocating time slots dynamically based on the presence of data. Time slots are assigned only when there is data to transmit, which optimizes the use of the communication channel.
### Applications of TDM
- **Telecommunications**: TDM is extensively used in telecommunication systems, such as in T1 and E1 lines, where multiple telephone calls are transmitted over a single line by assigning each call to a specific time slot.
- **Digital Audio and Video Broadcasting**: TDM is used in broadcasting systems to transmit multiple audio or video streams over a single channel, ensuring efficient use of bandwidth.
- **Computer Networks**: TDM is used in network protocols and systems to manage the transmission of data from multiple sources over a single network medium.
### Advantages of TDM
- **Efficient Use of Bandwidth**: TDM all
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356 Grid Converters for Photovoltaic and Wind Power Systems
0 10 20 30 40
-150
-100
-50
0
50
100
150
t [ms]
v
abc
b
v
a
v
c
v
b
V
a
V
c
V
(b)
(a)
148.7 3.6º
67.5 61.5º
112.8 165.5º
a
b
c
V
V
V
= ∠
= ∠ −
= ∠
Figure A.1 Unbalanced three-phase system: (a) instantaneous voltage waveforms and (b) phase voltage
phasors
The steady-state voltage waveforms of a three-phase unbalanced system together with their
phasor representation on a Gauss plane are shown in Figure A.1.
Applying the symmetrical components method, these three unbalanced phasors representing
the three-phase voltages can be transformed into a new set of three phasors representing the se-
quence components of one of the phases of the three-phase system. For example, the positive-,
negative- and zero-sequence phasors of phase a (
V +
a ,
V −
a and
V 0
a ) can be calculated by the
following transformation matrix:
V+−0(a) = [T+−0] Vabc (A.1)
with
Vabc =
⎡
⎢
⎣
Va
Vb
Vc
⎤
⎥
⎦ =
⎡
⎢
⎣
Va∠θa
Vb∠θb
Vc∠θc
⎤
⎥
⎦ , V+−0(a) =
⎡
⎢
⎣
V +
a
V −
a
V 0
a
⎤
⎥
⎦ =
⎡
⎢
⎣
V +
a ∠θ+
a
V −
a ∠θ−
a
V 0
a ∠θ0
a
⎤
⎥
⎦
[T+−0] = 1
3
⎡
⎢
⎣
1 α α2
1 α2
α
1 1 1
⎤
⎥
⎦ (A.2)
where α = ej2π/3
= 1∠120◦
is known as the Fortescue operator. The phasors representing the
sequence components for phases b and c are given by
V +
b = α2
V +
a ;
V −
b = α
V −
a
V +
c = α
V +
a ;
V −
c = α2
V −
a
(A.3)
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Appendix A: Space Vector Transformations of Three-Phase Systems 357
The inverse transformation to pass from the phasors representing symmetrical components of
phase a to the phasors representing the unbalanced phase voltages is given by
Vabc = [T+−0]−1
V+−0(a) (A.4)
with
[T+−0]−1
=
⎡
⎣
1 1 1
α2
α 1
α α2
1
⎤
⎦ (A.5)
As an example, the phasors representing the sequence components of the unbalanced voltages
of Figure A.1, together with their corresponding instantaneous waveforms, are shown in
Figure A.2.
A.3 Symmetrical Components in the Time Domain
Lyon extended the work of Fortescue and applied the method of the symmetrical components
in the time domain [2]. When the Fortescue transformation matrix of (A.2) is applied to the
following set of three-phase unbalanced sinusoidal waveforms:
vabc =
⎡
⎣
va
vb
vc
⎤
⎦ = v+
abc + v−
abc + v0
abc
= V +
⎡
⎢
⎣
cos(ωt)
cos(ωt − 2π
3
)
cos(ωt + 2π
3
)
⎤
⎥
⎦ + V −
⎡
⎢
⎣
cos(ωt)
cos(ωt + 2π
3
)
cos(ωt − 2π
3
)
⎤
⎥
⎦ + V 0
⎡
⎢
⎣
cos(ωt)
cos(ωt)
cos(ωt)
⎤
⎥
⎦ (A.6)
the resultant instantaneous variables are given by
v+−0 = [T+−0] vabc (A.7)
v+−0 =
⎡
⎢
⎣
v+
v−
v0
⎤
⎥
⎦ =
⎡
⎢
⎣
1
2
V +
ejωt
+ 1
2
V −
e− jωt
1
2
V +
e− jωt
+ 1
2
V −
ejωt
V 0
cos(ωt)
⎤
⎥
⎦ (A.8)
It is worth mentioning that the Lyon transformation is usually defined by the following
normalized matrix:
T
+−0
=
√
3 [T+−0] =
1
√
3
⎡
⎣
1 α α2
1 α2
α
1 1 1
⎤
⎦ (A.9)
where
T
+−0
−1
=
T
+−0
T
.
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358 Grid Converters for Photovoltaic and Wind Power Systems
(a)
(b)
(c)
0 10 20 30 40
-150
-100
-50
0
50
100
150
t [ms]
v
abc a
v+
b
v+
c
v+
a
V +
c
V +
b
V + 100 30º
100 90º
100 210º
a
b
c
V
V
V
+
+
+
= ∠
= ∠ −
= ∠ −
0 10 20 30 40
-150
-100
-50
0
50
100
150
t [ms]
v
abc
a
v−
c
v−
b
v− b
V −
a
V −
c
V −
50 40º
50 80º
50 160º
a
b
c
V
V
V
−
−
−
= ∠ −
= ∠
= ∠ −
0 10 20 30 40
-150
-100
-50
0
50
100
150
t [ms]
v
abc
0 0 0
a b c
v v v
= =
0 0 0
, ,
a b c
V V V
0
0
0
25 20º
25 20º
25 20º
a
b
c
V
V
V
= ∠ −
= ∠ −
= ∠ −
Figure A.2 Sequence components of the unbalanced three-phase system of Figure A.1: (a) positive-
sequence phasors, (b) negative-sequence phasors and (c) zero-sequence phasors
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Appendix A: Space Vector Transformations of Three-Phase Systems 359
in
v
3
1
2
in
v
α
( )
2
2
( ) in
in
LPF s
s
ω
ω
=
+
LPF
Figure A.3 Simple implementation of the a operator in the time domain
From (A.8) it can concluded that, independently of the scale factor used in the Lyon
transformation, the resulting vector consists of two complex elements,
v+
and
v−
, plus a
real element v0
. The complex elements
v+
and
v−
can be represented by instantaneous space
vectors, having the same amplitude and rotating in opposite directions. Therefore,
v+
and
v−
should not be mistaken for the positive- and negative-sequence voltage vectors v+
abc and
v−
abc. The real element v0
is directly related to the zero-sequence component of the original
three-phase voltage vector.
To calculate the positive- and negative-sequence voltage vectors, v+
abc and v−
abc, from the
unbalanced input vector vabc, it is necessary to translate the Fortescue operator, α, from
the frequency domain to the time domain. This translation can be performed by using a
simple time-shifting operator, provided the frequency of the sinusoidal input is a well-known
magnitude. In such a case, a 2
/3T time-shifted sinusoidal signal, with T the signal period, can
be understood as a 120◦
leaded version of the original sinusoidal signal. This operator in the
time domain is named a in this appendix. Since α = −1 2 + j
√
3 2, the a operator can be
implemented by using a proper filter to generate the 90◦
phase-shifting associated to the j
operator [3]. As an example, Figure A.3 shows a simple implementation of the a operator
based on a second-order low-pass filter (LPF) tuned at the input frequency with a damping
factor ξ = 1. The a2
operator can be implemented by multiplying the output signal of the LPF
by −1.
Therefore, the following expressions can be used to calculate the instantaneous positive-
and negative-sequence components of vabc:
v+
abc = [T+] vabc;
⎡
⎢
⎣
v+
a
v+
b
v+
c
⎤
⎥
⎦ =
1
3
⎡
⎢
⎣
1 a a2
a2
1 a
a a2
1
⎤
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.10)
v−
abc = [T−] vabc;
⎡
⎢
⎣
v−
a
v−
b
v−
c
⎤
⎥
⎦ =
1
3
⎡
⎢
⎣
1 a2
a
a 1 a2
a2
a 1
⎤
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.11)
A.4 Components αβ0 on the Stationary Reference Frame
Since the complex elements
v+
and
v−
in (A.8) are not independent from each other, only
three independent real components can be found among the elements resulting from the
transformation of (A.7). Therefore, a possible set of independent elements can be defined
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360 Grid Converters for Photovoltaic and Wind Power Systems
from (A.8) as
v+
,
v+
, v0
, although other combinations are also possible. At this
point, the following real transformation matrix can be written:
⎡
⎢
⎣
(
v+
)
(
v+
)
v0
⎤
⎥
⎦ =
1
3
⎡
⎢
⎣
1 (α) (α2
)
0 (α) (α2
)
1 1 1
⎤
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.12)
A similar reasoning caused Clarke to reformulate the Lyon transformation and to propose the
following transformation matrix [4]:
vαβ0 =
Tαβ0
vabc (A.13)
⎡
⎢
⎣
vα
vβ
v0
⎤
⎥
⎦ =
2
3
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 −
1
2
−
1
2
0
√
3
2
−
√
3
2
1
√
2
1
√
2
1
√
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.14)
where
Tαβ0
−1
=
Tαβ0
T
.
It is worth remarking here that the input and output vectors have the same norm when the
normalized transformation of (A.14) is applied, i.e.
v2
α + v2
β + v2
0 = v2
a + v2
b + v2
c (A.15)
As a consequence, when the normalized transformation of (A.14) is applied to the voltages
and currents of a three-phase system, power calculations will give rise to equivalent results for
both the abc and the αβ0 reference frames, i.e.
p = vαβ0 · iαβ0 = vabc · iabc (A.16)
The transformation
Tαβ0
can be rescaled as shown in the following equation when the
amplitude of the sinusoidal signals on the abc and the αβ0 reference frames should be equal,
i.e. when V̂α = V̂a:
⎡
⎢
⎣
vα
vβ
v0
⎤
⎥
⎦ =
2
3
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 −
1
2
−
1
2
0
√
3
2
−
√
3
2
1
√
2
1
√
2
1
√
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎣
va
vb
vc
⎤
⎥
⎦ (A.17)
The αβ0 reference frame is graphically depicted in Figure A.4. In this figure, the αβ plane holds
all the symmetrical vectors, i.e. those vαβ vectors with no zero sequence (va + vb + vc = 0),
whereas the 0 axis is aligned with the space diagonal of the abc reference frame.
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Appendix A: Space Vector Transformations of Three-Phase Systems 361
α
β
0
0
v
α
v
β
v
plane
α β
−
0 abc
αβ
= =
v v v
αβ
v
a
b
c
Figure A.4 Graphical representation of the αβ0 reference frame
A.5 Components dq0 on the Synchronous Reference Frame
Any voltage vector rotating on the αβ plane can be expressed on a synchronous reference frame
by using the Park transformation [5]. As depicted in Figure A.5, the synchronous reference
frame, also known as the dq reference frame, is based on two orthogonal dq axes, rotating at
frequency ω, which are placed at the θ = ω t angular position on the αβ plane. Thanks to its
rotating character, this transformation has been extensively used in the analysis of electrical
machines.
The transformation matrix to translate a voltage vector from the αβ0 stationary reference
frame to the dq0 synchronous reference frame is given by
vdq0 =
Tdq0
vαβ0 (A.18)
d
v
q
v
α
β
q
d
ω
α
v
β
v
θ
plane
α β
−
0
0
v
0 0
dq abc
αβ
= = =
v v v v
αβ
v
Figure A.5 Graphical representation of the dq0 reference frame
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362 Grid Converters for Photovoltaic and Wind Power Systems
⎡
⎣
vd
vq
v0
⎤
⎦ =
⎡
⎣
cos(θ) sin(θ) 0
− sin(θ) cos(θ) 0
0 0 1
⎤
⎦
⎡
⎣
vα
vβ
v0
⎤
⎦ (A.19)
where
Tdq0
−1
=
Tdq0
T
. Therefore, the transformation matrix to translate a voltage vector
from the abc stationary reference frame to the dq0 synchronous reference frame is given by
vdq0 = [Tθ ] vabc (A.20)
⎡
⎣
vd
vq
v0
⎤
⎦ =
2
3
⎡
⎢
⎢
⎢
⎢
⎣
cos(θ) cos(θ − 2π
3
) cos(θ + 2π
3
)
− sin(θ) − sin(θ − 2π
3
) − sin(θ + 2π
3
)
1
√
2
1
√
2
1
√
2
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎣
va
vb
vc
⎤
⎦ (A.21)
where [Tθ ]−1
= [Tθ ]T
.
The normalized transformations shown in (A.14) and (A.21) allow the norm of the voltage
vector to be conserved in all the reference frames; thus
v2
d + v2
q + v2
0 = v2
α + v2
β + v2
0 = v2
a + v2
b + v2
c (A.22)
Expressing voltage and currents using space vectors allows instantaneous phenomena in three-
phase systems to be studied using an efficient and elegant formulation. This formulation is
particularly useful to control active and reactive power components in three-phase systems.
References
[1] Fortescue, C. L. ‘Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks’. Transac-
tions of the AIEE, Part II, 37, 1918, 1027–1140.
[2] Lyon, W. V., Application of the Method of Symmetrical Components, New York: McGraw-Hill, 1937.
[3] Iravani, M. R. and Karimi-Ghartemani, M., ‘Online Estimation of Steady State and Instantaneous Symmetrical
Components’. Proceedings of the IEE on Generation, Transmission and Distribution, 150(5), September 2003,
616–622.
[4] Clarke, E., Circuit Analysis of AC Power Systems, Vol. 1, New York: John Wiley Sons, Inc., 1950.
[5] Park, R. H., ‘Two Reaction Theory of Synchronous Machines. Generalized Method of Analysis – Part I’. In
Proceedings of the Winter Convention of the AIEE, 1929, pp. 716–730.