This document is Swabhiman Singh Parida's dissertation submitted in partial fulfillment of the requirements for a Bachelor of Technology degree in Electrical Engineering. It discusses network topology and graph theory. The dissertation includes declarations signed by Parida and his supervisor Durga Prasanna Mohanty. It also provides acknowledgements and contains chapters on topics like circuit elements and laws, network analysis, network theorems, and different types of network topologies and graphs.
This document provides an overview of circuit theory concepts including:
- Electric circuits are interconnections of electrical elements.
- Charge is the most basic quantity and is measured in coulombs. Current is the rate of charge flow measured in amperes.
- Voltage is the energy required to move a unit charge through a circuit element and is measured in volts.
- Power is the rate of energy use/production and is measured in watts.
- Circuit elements include passive (resistors, capacitors, inductors) and active (sources) components. Kirchhoff's laws and Ohm's law govern circuit analysis.
- Nodal and mesh analysis provide systematic techniques for analyzing circuits by
1) The document discusses DC fundamentals and circuits, covering topics like charge, current, voltage, power, energy, Ohm's law, and Kirchhoff's laws. It also covers basic circuit analysis using these principles.
2) Key concepts discussed include the definitions of current, voltage, resistance, and time constants. Kirchhoff's laws and Ohm's law are also summarized.
3) Examples are provided to demonstrate using these principles to solve circuits for unknown currents and voltages. Circuit analysis techniques like mesh current analysis and nodal voltage analysis are also mentioned.
Ekeeda Provides Online Electrical and 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/Electrical-and-Electronics-Engineering
Electrical Engineering is the Branch of Engineering. Electrical Engineering field requires an understanding of core areas including Thermal and Hydraulics Prime Movers, Analog Electronic Circuits, Network Analysis and Synthesis, DC Machines and Transformers, Digital Electronic Circuits, Fundamentals of Power Electronics, Control System Engineering, Engineering Electromagnetics, Microprocessor and Microcontroller. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus. Visit : https://ekeeda.com/streamdetails/stream/Electrical-and-Electronics-Engineering
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 discusses circuit and network theory. It covers topics such as circuit elements and laws, magnetic circuits, network analysis, network theorems, AC circuits and resonance, coupled circuits, transients, two-port networks, and filters. Mesh analysis is introduced as a technique for network analysis that is applicable to planar networks containing voltage sources. The key steps are selecting mesh currents, then writing and solving KVL equations in terms of the unknown currents.
This document provides an overview of Circuit Theory (EE102) Lecture 1, covering basic concepts in electric circuits including:
- Systems of units used to measure electric properties like current and voltage.
- Basic circuit elements like resistors, sources, and nodes and branches.
- Kirchhoff's laws and techniques for analyzing series and parallel circuits.
- Transformations between wye and delta networks.
Worked examples are provided to illustrate applying concepts like Ohm's law, Kirchhoff's laws, and calculating equivalent resistances for series and parallel circuits.
This document provides an overview of circuit theory concepts including:
- Electric circuits are interconnections of electrical elements.
- Charge is the most basic quantity and is measured in coulombs. Current is the rate of charge flow measured in amperes.
- Voltage is the energy required to move a unit charge through a circuit element and is measured in volts.
- Power is the rate of energy use/production and is measured in watts.
- Circuit elements include passive (resistors, capacitors, inductors) and active (sources) components. Kirchhoff's laws and Ohm's law govern circuit analysis.
- Nodal and mesh analysis provide systematic techniques for analyzing circuits by
1) The document discusses DC fundamentals and circuits, covering topics like charge, current, voltage, power, energy, Ohm's law, and Kirchhoff's laws. It also covers basic circuit analysis using these principles.
2) Key concepts discussed include the definitions of current, voltage, resistance, and time constants. Kirchhoff's laws and Ohm's law are also summarized.
3) Examples are provided to demonstrate using these principles to solve circuits for unknown currents and voltages. Circuit analysis techniques like mesh current analysis and nodal voltage analysis are also mentioned.
Ekeeda Provides Online Electrical and 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/Electrical-and-Electronics-Engineering
Electrical Engineering is the Branch of Engineering. Electrical Engineering field requires an understanding of core areas including Thermal and Hydraulics Prime Movers, Analog Electronic Circuits, Network Analysis and Synthesis, DC Machines and Transformers, Digital Electronic Circuits, Fundamentals of Power Electronics, Control System Engineering, Engineering Electromagnetics, Microprocessor and Microcontroller. Ekeeda offers Online Mechanical Engineering Courses for all the Subjects as per the Syllabus. Visit : https://ekeeda.com/streamdetails/stream/Electrical-and-Electronics-Engineering
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 discusses circuit and network theory. It covers topics such as circuit elements and laws, magnetic circuits, network analysis, network theorems, AC circuits and resonance, coupled circuits, transients, two-port networks, and filters. Mesh analysis is introduced as a technique for network analysis that is applicable to planar networks containing voltage sources. The key steps are selecting mesh currents, then writing and solving KVL equations in terms of the unknown currents.
This document provides an overview of Circuit Theory (EE102) Lecture 1, covering basic concepts in electric circuits including:
- Systems of units used to measure electric properties like current and voltage.
- Basic circuit elements like resistors, sources, and nodes and branches.
- Kirchhoff's laws and techniques for analyzing series and parallel circuits.
- Transformations between wye and delta networks.
Worked examples are provided to illustrate applying concepts like Ohm's law, Kirchhoff's laws, and calculating equivalent resistances for series and parallel circuits.
This document covers fundamental circuit analysis concepts including:
1) Ohm's law defines the relationship between current, voltage, and resistance in a circuit. Kirchoff's laws (KVL and KCL) are also introduced.
2) Series and parallel resistor combinations are examined along with voltage and current division techniques.
3) Wye-delta transformations allow the analysis of resistor networks that are neither purely series nor parallel.
ELECTRICAL AND INSTRUMENTATION ENGINEERING COURSE FOR OIL AND GAS FACILITIES
Please subscribe to my YouTube Channel for best training lectures:
https://www.youtube.com/channel/UCRkUJFOsyZG1E1LDWzUr_hw
The document discusses various circuit theorems including:
1. Linearity property and superposition principle which allow complex circuits to be simplified by treating sources individually.
2. Source transformations allow replacing voltage sources in series with resistances by current sources in parallel with resistances.
3. Thevenin's theorem states any linear two-terminal circuit can be reduced to a voltage source in series with a resistance.
4. Examples are provided to demonstrate applying these theorems to solve for unknown voltages and currents.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
This document provides an overview of direct current (DC) circuits and circuit analysis techniques. It defines key concepts like voltage sources, current sources, ideal and real sources, and dependent and independent sources. It also explains Kirchhoff's laws, nodal analysis, and mesh analysis. Kirchhoff's current law states that the algebraic sum of currents at a node is zero. Kirchhoff's voltage law states that the algebraic sum of voltages in a closed loop is zero. Nodal analysis uses Kirchhoff's current law to set up equations relating node voltages. Mesh analysis uses Kirchhoff's voltage law to set up equations relating mesh currents.
Menuntut ilmu adalah TAQWA - Seeking knowledge is piety.
Menyampaikan ilmu adalah IBADAH - Conveying knowledge is worship.
Mengulang-ulang ilmu adalah ZIKIR - Repeating knowledge is remembrance of God.
Mencari ilmu adalah JIHAD - Seeking knowledge is jihad.
Network theorems for electrical engineeringKamil Hussain
The document discusses several circuit analysis theorems and methods. Kirchhoff's laws describe the conservation of charge and energy in circuits. Mesh analysis and nodal analysis are methods to solve circuits by assigning currents or voltages and setting up equations based on Kirchhoff's laws. The superposition theorem allows analyzing circuits with multiple sources by solving for each source independently and summing the results.
This chapter provides complete solution of of first, Second order differential equations of series & parallel R-L, R-C, R-L-C circuits, bu using different methods.
1) The document provides an introduction and outlines for a circuit analysis course. It discusses topics like resistance, current, voltage, and units of measurement.
2) It defines electrical resistance as the opposition to electron flow in a conductor. Resistance depends on the material's resistivity, length, and cross-sectional area. It also describes different types of resistors.
3) Formulas are provided for calculating resistance of circular wires as well as the relationship between resistance and conductance. Color coding schemes for fixed resistors are also explained.
The document discusses operational amplifiers (op amps) and their applications in different circuit configurations:
1) An op amp is an electronic device that can perform mathematical operations like addition, subtraction, etc. It has high gain, very high input impedance, and very low output impedance.
2) Common op amp circuit configurations include the inverting amplifier, non-inverting amplifier, summing amplifier, difference amplifier, and instrumentation amplifier.
3) The summing amplifier produces an output voltage that is the weighted sum of its input voltages. The difference amplifier amplifies the difference between its two input voltages and rejects any components that are common to both inputs.
Ekeeda - First Year Enginering - Basic Electrical EngineeringEkeedaPvtLtd
The First Year engineering course seems more like an extension of the subjects that students have learned in their 12th class. Subjects like Engineering Physics, Chemistry, and Mathematics, are incorporated into the curriculum. Students will learn about some of the engineering subjects in this first year, and these subjects are similar to all the branches. Everyone will learn some basics related to the other streams in their first year. Ekeeda offers Online First Year Engineering Courses for all the Subjects as per the Syllabus.
The document summarizes key concepts about Kirchhoff's laws, Thévenin's and Norton's theorems, and network analysis techniques. Specifically:
- Kirchhoff's laws deal with current and voltage in electrical circuits and are based on conservation of charge and energy. The junction rule states the sum of currents at a node is zero, and the loop rule states the algebraic sum of voltages in a closed loop is zero.
- Thévenin's and Norton's theorems allow any two-terminal linear network to be reduced to an equivalent circuit with a voltage or current source and single impedance. This simplifies analysis and understanding how the network responds to changes.
- Network analysis methods like
This document provides an introduction to basic electrical concepts including charge, current, voltage, power, energy, and circuit elements. It defines the international system of units used in electrical engineering. Key concepts covered include defining the ampere as the unit of current representing the flow of electric charge, defining voltage as the work required to move a unit of charge from one point to another, and defining power as the rate at which energy is transferred. Circuit analysis techniques are introduced for studying the behavior of electric circuits.
This course introduces the basic concepts of circuit analysis which is the foundation for electrical engineering. It covers topics like Kirchhoff's laws, network reduction techniques, nodal analysis, mesh analysis, AC circuits, magnetic circuits, resonance, network topology, and network theorems. The objective is to provide students with a thorough understanding of circuit analysis concepts that can be applied to real-world problems. The course is divided into five units covering these essential circuit analysis topics.
Basic Electrical Engineering Module 1 Part 1Divya15121983
This document provides an overview of basic electrical engineering concepts including Ohm's Law, series and parallel circuits, and Kirchhoff's Laws. It defines Ohm's Law as stating that current is directly proportional to voltage and inversely proportional to resistance. Kirchhoff's Current Law and Voltage Law are introduced as the principles that the algebraic sum of currents at a junction is zero and the algebraic sum of voltages around a closed loop is also zero. An example circuit problem is worked through using these laws to solve for unknown currents.
An electric circuit is a path in which electrons from a voltage or current source flow. The point where those electrons enter an electrical circuit is called the "source" of electrons.
This presentation provides an explanation of Active & Passive Circuit Element: Independent & dependent voltage & current sources, R, L, C, and Their mathematical modes, Voltage current power relations, Series and Parallel circuits, Kirchhoff's Laws. It also provides an information about
Classification of elements with numerical examples.
1. Kirchhoff's laws (KCL and KVL) form the foundation for electric circuit analysis. KCL states that the algebraic sum of currents entering a node is zero, while KVL states that the algebraic sum of voltages around a closed loop is zero.
2. A node is the point of connection between two or more branches, while a loop is any closed path in a circuit without repeating nodes. Kirchhoff's laws can be applied to nodes, closed surfaces, and loops.
3. Ohm's law defines the relationship between current and voltage in a resistor as V=IR. Resistors can be connected in series or parallel, and equivalent resistances can be calculated using their individual
A circuit consists of electrical elements connected in a closed loop to allow current flow. Key concepts include:
- Current is the flow of electric charge. Voltage is electrical potential difference and power is the rate of work done.
- Circuits have active elements like voltage and current sources that supply energy and passive elements like resistors, inductors and capacitors that receive energy.
- Kirchhoff's laws state that the algebraic sum of voltages around any loop is zero and the algebraic sum of currents at any node is zero.
- Resistors in series add, resistors in parallel calculate using reciprocal formula. Source transformations allow representing one source type as another while maintaining terminal characteristics.
These slides explain the topics mentioned in Chapter 1, part (a) of the course EE110-Basic Electrical and Electronics Engineering, prescribed for non-circuit branches of engineering at JSS Science & Technology University, Sri Jayachamarajendra College of Engineering, Mysuru, India
This document covers fundamental circuit analysis concepts including:
1) Ohm's law defines the relationship between current, voltage, and resistance in a circuit. Kirchoff's laws (KVL and KCL) are also introduced.
2) Series and parallel resistor combinations are examined along with voltage and current division techniques.
3) Wye-delta transformations allow the analysis of resistor networks that are neither purely series nor parallel.
ELECTRICAL AND INSTRUMENTATION ENGINEERING COURSE FOR OIL AND GAS FACILITIES
Please subscribe to my YouTube Channel for best training lectures:
https://www.youtube.com/channel/UCRkUJFOsyZG1E1LDWzUr_hw
The document discusses various circuit theorems including:
1. Linearity property and superposition principle which allow complex circuits to be simplified by treating sources individually.
2. Source transformations allow replacing voltage sources in series with resistances by current sources in parallel with resistances.
3. Thevenin's theorem states any linear two-terminal circuit can be reduced to a voltage source in series with a resistance.
4. Examples are provided to demonstrate applying these theorems to solve for unknown voltages and currents.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
This document provides an overview of direct current (DC) circuits and circuit analysis techniques. It defines key concepts like voltage sources, current sources, ideal and real sources, and dependent and independent sources. It also explains Kirchhoff's laws, nodal analysis, and mesh analysis. Kirchhoff's current law states that the algebraic sum of currents at a node is zero. Kirchhoff's voltage law states that the algebraic sum of voltages in a closed loop is zero. Nodal analysis uses Kirchhoff's current law to set up equations relating node voltages. Mesh analysis uses Kirchhoff's voltage law to set up equations relating mesh currents.
Menuntut ilmu adalah TAQWA - Seeking knowledge is piety.
Menyampaikan ilmu adalah IBADAH - Conveying knowledge is worship.
Mengulang-ulang ilmu adalah ZIKIR - Repeating knowledge is remembrance of God.
Mencari ilmu adalah JIHAD - Seeking knowledge is jihad.
Network theorems for electrical engineeringKamil Hussain
The document discusses several circuit analysis theorems and methods. Kirchhoff's laws describe the conservation of charge and energy in circuits. Mesh analysis and nodal analysis are methods to solve circuits by assigning currents or voltages and setting up equations based on Kirchhoff's laws. The superposition theorem allows analyzing circuits with multiple sources by solving for each source independently and summing the results.
This chapter provides complete solution of of first, Second order differential equations of series & parallel R-L, R-C, R-L-C circuits, bu using different methods.
1) The document provides an introduction and outlines for a circuit analysis course. It discusses topics like resistance, current, voltage, and units of measurement.
2) It defines electrical resistance as the opposition to electron flow in a conductor. Resistance depends on the material's resistivity, length, and cross-sectional area. It also describes different types of resistors.
3) Formulas are provided for calculating resistance of circular wires as well as the relationship between resistance and conductance. Color coding schemes for fixed resistors are also explained.
The document discusses operational amplifiers (op amps) and their applications in different circuit configurations:
1) An op amp is an electronic device that can perform mathematical operations like addition, subtraction, etc. It has high gain, very high input impedance, and very low output impedance.
2) Common op amp circuit configurations include the inverting amplifier, non-inverting amplifier, summing amplifier, difference amplifier, and instrumentation amplifier.
3) The summing amplifier produces an output voltage that is the weighted sum of its input voltages. The difference amplifier amplifies the difference between its two input voltages and rejects any components that are common to both inputs.
Ekeeda - First Year Enginering - Basic Electrical EngineeringEkeedaPvtLtd
The First Year engineering course seems more like an extension of the subjects that students have learned in their 12th class. Subjects like Engineering Physics, Chemistry, and Mathematics, are incorporated into the curriculum. Students will learn about some of the engineering subjects in this first year, and these subjects are similar to all the branches. Everyone will learn some basics related to the other streams in their first year. Ekeeda offers Online First Year Engineering Courses for all the Subjects as per the Syllabus.
The document summarizes key concepts about Kirchhoff's laws, Thévenin's and Norton's theorems, and network analysis techniques. Specifically:
- Kirchhoff's laws deal with current and voltage in electrical circuits and are based on conservation of charge and energy. The junction rule states the sum of currents at a node is zero, and the loop rule states the algebraic sum of voltages in a closed loop is zero.
- Thévenin's and Norton's theorems allow any two-terminal linear network to be reduced to an equivalent circuit with a voltage or current source and single impedance. This simplifies analysis and understanding how the network responds to changes.
- Network analysis methods like
This document provides an introduction to basic electrical concepts including charge, current, voltage, power, energy, and circuit elements. It defines the international system of units used in electrical engineering. Key concepts covered include defining the ampere as the unit of current representing the flow of electric charge, defining voltage as the work required to move a unit of charge from one point to another, and defining power as the rate at which energy is transferred. Circuit analysis techniques are introduced for studying the behavior of electric circuits.
This course introduces the basic concepts of circuit analysis which is the foundation for electrical engineering. It covers topics like Kirchhoff's laws, network reduction techniques, nodal analysis, mesh analysis, AC circuits, magnetic circuits, resonance, network topology, and network theorems. The objective is to provide students with a thorough understanding of circuit analysis concepts that can be applied to real-world problems. The course is divided into five units covering these essential circuit analysis topics.
Basic Electrical Engineering Module 1 Part 1Divya15121983
This document provides an overview of basic electrical engineering concepts including Ohm's Law, series and parallel circuits, and Kirchhoff's Laws. It defines Ohm's Law as stating that current is directly proportional to voltage and inversely proportional to resistance. Kirchhoff's Current Law and Voltage Law are introduced as the principles that the algebraic sum of currents at a junction is zero and the algebraic sum of voltages around a closed loop is also zero. An example circuit problem is worked through using these laws to solve for unknown currents.
An electric circuit is a path in which electrons from a voltage or current source flow. The point where those electrons enter an electrical circuit is called the "source" of electrons.
This presentation provides an explanation of Active & Passive Circuit Element: Independent & dependent voltage & current sources, R, L, C, and Their mathematical modes, Voltage current power relations, Series and Parallel circuits, Kirchhoff's Laws. It also provides an information about
Classification of elements with numerical examples.
1. Kirchhoff's laws (KCL and KVL) form the foundation for electric circuit analysis. KCL states that the algebraic sum of currents entering a node is zero, while KVL states that the algebraic sum of voltages around a closed loop is zero.
2. A node is the point of connection between two or more branches, while a loop is any closed path in a circuit without repeating nodes. Kirchhoff's laws can be applied to nodes, closed surfaces, and loops.
3. Ohm's law defines the relationship between current and voltage in a resistor as V=IR. Resistors can be connected in series or parallel, and equivalent resistances can be calculated using their individual
A circuit consists of electrical elements connected in a closed loop to allow current flow. Key concepts include:
- Current is the flow of electric charge. Voltage is electrical potential difference and power is the rate of work done.
- Circuits have active elements like voltage and current sources that supply energy and passive elements like resistors, inductors and capacitors that receive energy.
- Kirchhoff's laws state that the algebraic sum of voltages around any loop is zero and the algebraic sum of currents at any node is zero.
- Resistors in series add, resistors in parallel calculate using reciprocal formula. Source transformations allow representing one source type as another while maintaining terminal characteristics.
These slides explain the topics mentioned in Chapter 1, part (a) of the course EE110-Basic Electrical and Electronics Engineering, prescribed for non-circuit branches of engineering at JSS Science & Technology University, Sri Jayachamarajendra College of Engineering, Mysuru, India
Ekeeda Provides Online Video Lectures, Tutorials & Engineering Courses Available for Top-Tier Universities in India. Lectures from Highly Trained & Experienced Faculty!
This document outlines the contents and objectives of an electrical circuits course. It is divided into 12 lectures covering topics such as introduction to electrical circuits, DC machines, AC machines, transformers, rectifiers, integrated circuits, and transistors. The objectives of the course are to introduce students to basic concepts of electrical and electronic circuits, analyze circuits, and understand DC motors, AC motors, transformers, rectifiers, and transistors. It also lists the textbook that will be used and provides sample lecture content, including defining circuit elements and analysis techniques like Kirchhoff's laws.
The document provides an overview of topics related to electrical circuits and electromagnetism including:
1) Definitions of key circuit elements and analysis techniques like Kirchhoff's laws, superposition theorem, and Thevenin's theorem.
2) Concepts in electromagnetism including Biot-Savart law, Ampere's law, Faraday's law, and magnetic circuits.
3) Analysis of AC circuits including waveform properties, phasor representation, and resonance in RLC circuits.
This document provides information about Parshva Classes, an educational institution that offers courses for engineering degrees and diplomas from various universities. It lists the contact details and addresses of the institution's main and branch offices. The bulk of the document consists of definitions and explanations of various electrical circuit and network terms like active and passive elements, nodes, branches, loops, meshes, Kirchoff's laws, Maxwell's loop theorem, Thevenin's theorem and Norton's theorem. Key concepts from circuit analysis such as superposition theorem and reciprocity theorem are also summarized.
1. The document discusses Ohm's law and basic electrical circuit concepts such as resistance, capacitance, inductance, and power.
2. It introduces modern electron theory and defines an atom as consisting of a positively charged nucleus surrounded by negatively charged electrons.
3. Key circuit elements like resistors, capacitors, and inductors are defined in terms of how they store or dissipate electrical energy. Kirchhoff's laws and techniques for analyzing circuits like source transformations are also summarized.
The document discusses key concepts in electric circuits including:
1. An electric circuit is a continuous conducting path for electric current to flow, consisting of wires and other resistance like a bulb connected between the terminals of a battery or cell.
2. Electric current is the flow of electric charge, usually carried by moving electrons in a wire, and is measured in Amperes.
3. Potential difference (PD) is the work done to move a unit charge between two points in a circuit, measured in Volts using a voltmeter.
4. Ohm's law states the current through a conductor is directly proportional to the PD across it for a given temperature.
Electricity and its uses were presented. Electricity flows as an electric current from the positive terminal to the negative terminal of a circuit. Current is measured in Amperes and is proportional to the amount of charge flowing across a conductor per unit time. There are two types of electric charge: positive and negative. Opposite charges attract while like charges repel. Electric circuits use symbols to represent components and diagrams to show how components are connected. Ohm's law states that the current through a conductor is directly proportional to the potential difference across it. Resistance opposes the flow of current and can be measured in Ohms.
This document provides information about electricity and electrical circuits. It defines electric current as the flow of electrons through a conductor. It explains that electric current is measured in amperes and depends on the rate of flow of electric charge. It also defines electric potential difference, voltage, resistance, and Ohm's law. Additionally, it describes series and parallel circuits and how to calculate equivalent resistance. It discusses electrical energy, power, and how current produces heat when passed through a resistor.
This document provides an overview of circuit theory concepts including:
- Electric circuits are interconnections of electrical elements.
- Charge is the most basic quantity and is measured in coulombs. Current is the rate of charge flow measured in amperes.
- Voltage is the energy required to move a unit charge through a circuit element and is measured in volts.
- Power is the rate of energy use/production and is measured in watts.
- Circuit elements include passive (resistors, capacitors, inductors) and active (sources) components. Kirchhoff's laws and Ohm's law govern circuit analysis.
- Nodal and mesh analysis provide systematic techniques for analyzing circuits by
1. Electric current is the flow of electric charge, usually carried by moving electrons through a conductor. The direction of conventional current is opposite to the direction of electron flow.
2. An electric circuit is a closed path formed by conductors through which electric current can flow.
3. Resistance is a property of conductors that opposes the flow of electric current. The resistance of a conductor depends on its material, length, and cross-sectional area.
1) The document discusses DC and AC circuits, defining key concepts like voltage, current, resistance, inductance, and capacitance.
2) It describes different types of circuit elements and how they are connected in series, parallel and series-parallel configurations.
3) Kirchhoff's laws and theorems like superposition, phasor representation, and analysis of RL, RC, and RLC circuits under alternating current are explained.
Current and Electricity for class 10.pdfJackHassan2
Download our comprehensive Class 10 Current and Electricity PDF guide to master the fundamentals of electrical circuits, current flow, and key concepts in this critical science topic. This PDF provides clear explanations, diagrams, and practice questions to enhance your understanding and preparation. Covering the essential curriculum, it's an invaluable resource for students aiming for excellence in their science studies.
Current and Electricity for class 10.pdfJackHassan2
Download our comprehensive Class 10 Current and Electricity PDF guide to master the fundamentals of electrical circuits, current flow, and key concepts in this critical science topic. This PDF provides clear explanations, diagrams, and practice questions to enhance your understanding and preparation. Covering the essential curriculum, it's an invaluable resource for students aiming for excellence in their science studies.
Current electricity is a fundamental concept in physics, dealing with the flow of electric charge through a conductor. It forms the basis of various electrical devices and systems that we encounter in our daily lives.
For more information, visit-www.vavaclasses.com
The document discusses key concepts in electricity including electric current, electric circuits, potential difference, resistance, and Ohm's Law. It defines electric current as the flow of electrons through a conductor. An electric circuit is a continuous closed path for electric current to flow. Potential difference is the difference in electric potential needed to cause current flow. Ohm's Law states that current is directly proportional to potential difference in a conductor. Resistance depends on the material and dimensions of the conductor.
1. The document discusses electricity, including electric charge, current, potential difference, and circuits. It defines key terms and concepts and provides examples of calculations.
2. Series and parallel circuits are analyzed and compared. Equations for current, voltage, and resistance in each type of circuit are provided.
3. The relationship between potential difference and current is explored through Ohm's Law. Factors that affect resistance are also described.
This document provides a summary of key concepts in electricity including:
1. Electric current is the flow of electrons through a conductor measured in amperes. Current flows from the positive terminal to the negative terminal of a battery.
2. Potential difference is the difference in electric potential provided by a battery that causes electric current. It is measured in volts.
3. An electric circuit is a continuous loop through which electric current can flow, including components like batteries, wires, switches, and resistors.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Generative AI leverages algorithms to create various forms of content
Rec report
1. NETWORK TOPOLOGY AND GRAPH THEORY
Dissertation/ Seminar report submitted in partial fulfilment of the requirement for
the degree of
BACHELOR OF TECHNOLOGY
IN
ELECTRICAL ENGINEERING
By
Swabhiman Singh Parida
UNDER THE SUPERVISION OF
DURGA PRASANNA MOHANTY
HEAD OF DEPARTMENT
Rajdhani Engineering College, Bhubaneswar
2. DECLARATION
We hereby declare that the work reported in the B-Tech Seminar
Report entitled “NETWORK TOPOLOGY AND GRAPH THEORY”
submitted at, RAAJDHANI ENGINEERING COLLEGEBHUBANESWAR,
India, is an
authentic record of our work carried out under the supervision
of Durga Mohanty. We have not submitted this work elsewhere
for any other degree or diploma.
SWABHIMAN SINGH PARIDA
Department of ELECTRICAL ENGINEERING
Raajdhani Engineering College, Bhubaneswar
3. CERTIFICATE
This is to certify that the work reported in the B-Tech. Seminar Report entitled “Network
Topology and Graph Theory”, submitted by Swabhiman Singh Parida (1821294386) is an
authentic work applied by
him beneath my oversight and steerage for the partial fulfilment of the wants for the award of
Bachelor of Technology Degree in Electrical Engineering at RAAJDHANI ENGINEERING
COLLEGE, BHUBANESWAR.
To the most effective of my information, the matter embodied within the project has not been
submitted to the other University / Institute for the award of any Degree or certification.
(Durga Prasanna Mohanty)
Head of Department
Department of Electrical Engineering
Raajdhani Engineering College, Bhubaneswar, India
4. Acknowledgements
I express my profound gratitude and indebtedness to DURGA PRASANNA MOHANY,
Department of Electrical Engineering, REC, BBSR for introducing this topic and for his
ennobling intellectual support, valuable suggestion, esteemed guidelines and constant insight
overview throughout the project work.
Finally, I’m terribly appreciative to my Teacher who helped me in coming up
with the whole My Report.
I must acknowledge the educational resources that I actually have got from REC Bhubaneswar.
Date-
Bhubaneswar
Swabhiman Singh Parida
(Roll No: 1821294386)
5. Contents
o Circuit Element and Laws
o Network Analysis
o Network Theorems
o Topology Graph and its types
▪ Tree
▪ Twigs
• Co-tree
6. Circuit Elements and Laws
Voltage
Energy is required for the movement of charge from one point to
another. Let W Joules of energy be required to move positive charge Q
columbs from a point a to point b in a circuit. We say that a voltage exists
between the two points. The voltage V between two points may be
defined in terms of energy that would be required if a charge were
transferred from one point to the other. Thus, there can be a voltage
between two points even if no charge is actually moving from one to the
other. Voltage between a and b is given by
Hence Electric Potential (V) =
Worked are (W) in Joules
Ch arge (Q)in columbs
Current:
An electric current is the movement of electric charges along a definite path. In case of a
conductor the moving charges are electrons.
The unit of current is the ampere. The ampere is defined as that current which when
flowing in two infinitely long parallel conductors of negligible cross section, situated 1 meter
apart in Vacuum, produces between the conductors a force of 2 x 10-7
Newton per metre
length.
Power : Power is defined as the work done per unit time. If a field F newton acts for t seconds
through a distance d metres along a straight line, work done W = Fxd N.m. or J. The power
P, either generated or dissipated by the circuit element.
P =
w
=
F x d
t t
Power can also be written as Power = Work
time
=
Work
Charge
x
Ch arge
=
Time
Voltage x Current
P = V x I watt.
Energy: Electric energy W is defined as the Power Consumed in a given time. Hence, if current
IA flows in an element over a time period t second, when a voltage V volt is applied across it, the
energy consumed is given by
W = P x t = V x I x t J or watt. Second.
The unit of energy W is Joule (J) or watt. Second. However, in practice, the unit of energy is
kilowatt. Hour (Kwh).
7. Conductance : Voltage is induced in a stationary conductor when placed in a varying magnetic
field. The induced voltage I is proportional to the time rate of change of current, di/dt producing
the magnetic field.
Therefore e
di
dt
Or e = L
di
e and I are both function of time. The proportionality constant L is called inductance.
The Unit of inductance is Henry (H dt).
Capacitance : A capacitor is a Physical device, which when polarized by an electric field by
applying a suitable voltage across it, stores energy in the form of a charge separation.
The ability of the capacitor to store charge is measured in terms of capacitance.
Capacitence of a capacitor is defined as the charge stored per Volt applied.
C =
q
=
Coulomb
= Farad
v Volt
Active and passive Branch:
A branch is said to be active when it contains one or more energy sources. A passive branch
does not contain an energy source.
Branch: A branch is an element of the network having only two terminals.
Bilateral and unilateral element:
A bilateral element conducts equally well in either direction. Resistors and inductors are
examples of bilateral elements. When the current voltage relations are different for the two
directions of current flow, the element is said to be unilateral. Diode is a unilateral element.
Resistance: According to Ohm’s law potential difference (V) across the ends of a conductor is
proportional to the current (I) flowing through the conductor at a constant temperature.
Mathematically Ohm’s law is expressed as
Or R =
V I or V = R x I
V / I
where R is the proportionality constant and is designated as the conductor
resistance and has the unit of Ohm ().
8. Linear Elements: When the current and voltage relationship in an element can be simulated by
a linear equation either algebraic, differential or integral type, the element is said to be linear
element.
Non-Linear Elements: When the current and voltage relationship in an element cannot be
simulated by a linear equation, the element is said to be nonlinear elements.
Kirchhoff’s Voltage Law (KVL) :
The algebraic sum of Voltages (or voltage drops) in any closed path or loop is Zero.
Application of KVL with series connected voltage source:
Fig. 1.1
V1 + V2 – IR1 – IR2 = 0
= V1 + V2 = I (R1 + R2)
I =
V1 + V2
R1 + R2
Application of KVL while voltage sources are connected in opposite polarity.
9. Fig. 1.2
V1 – IR1 – V2 – IR2 – IR3 = 0
➢ V1 – V2 = IR1 + IR2 + IR3
➢ V1 – V2 = I (R1 + IR2 + IR3)
I = V1 − V2
2
R1 + R 2 + R
Kirchhoff’s Current Law (KCL) :
The algebraic sum of currents meeting at a junction or mode is zero.
Fig. 1.3
Considering five conductors, carrying currents I1, I2, I3, I4 and I5 meeting at a point O. Assuming
the incoming currents to be positive and outgoing currents negative.
I1 + (-I2) + I3 + (-I4) + I5 = 0 I1
– I2 + I3 – I4 + I5 = 0
I1 + I3 + I5 = I2 + I4
Thus, above Law can also be stated as the sum of currents flowing towards any junction in an
electric circuit is equal to the sum of the currents flowing away from that junction
Voltage Division (Series Circuit)
Considering a voltage source I with resistors R1 and R2 in series across it.
Fig. 1.4
10. Voltage drop across R1 = I. R1 =
I =
E.R1
E R1
+ R2
R1 + R 2
Similarly, voltage drop across R2 = I.R2 =
E.R 1
R1 + R 2
Current Division :
A parallel circuit acts as a current divider as the current divides in all branches in a parallel
circuit.
Fig. 1.5
Fig. shown the current I has been divided into I1 and I2 in two parallel branches with
resistances R1 and R2 while V is the voltage drop across R1 and R2.
I1 =
V
R1
an
d
I2 =
2
Let R = Total resistance of the circuit.
Hence
1
= =
1
+
1
R
➢ R =
R1 R2
R1R 2
R1 + R 2
R
V
R
11. V R1R2
R1 + R2
=
V(R1
+ R 2 )
R1
R
2
But = V = I1R1 = I2R2
R1R 2
➢ I = I1R1
R1
+R2
➢ I =
I1 (R1 + R 2 ) R
2
Therefore
Similarly it can be derived that
IR1
R1 + R 2
I2 =
IR2
R1 + R2
I1 =
12. c d
R4
p
R1 V3 R8
a b e
R2
K h g f
NETWORK ANALYSIS
Different terms are defined below:
1. Circuit: A circuit is a closed conducting path through which an electric current either
. flow or is intended flow
2. Network: A combination of various electric elements, connected in any manner.
Whatsoever, is called an electric network
3. Node: it is an equipotential point at which two or more circuit elements are joined.
4. Junction: it is that point of a network where three or more circuit elements are joined.
5. Branch: it is a part of a network which lies between junction points.
6. Loop: It is a closed path in a circuit in which no element or node is accounted more than
once.
7. Mesh: It is a loop that contains no other loop within it.
Example 3.1 In this circuit configuration of figure 3.1, obtain the no. of i) circuit elements ii)
nodes iii) junction points iv) branches and v) meshes.
R5
R6
V1 R7
13. Solution: i) no. of circuit elements = 12 (9 resistors + 3 voltage sources)
ii) no. of nodes =10 (a, b, c, d, e, f, g, h, k, p)
iii) no. of junction points =3 (b, e, h)
iv) no. of branches = 5 (bcde, be, bh, befgh, bakh)
v) no. of meshes = 3 (abhk, bcde, befh)
3.2 MESH ANALYSIS
Mesh and nodal analysis are two basic important techniques used in finding solutions
for a network. The suitability of either mesh or nodal analysis to a particular problem depends
mainly on the number of voltage sources or current sources .If a network has a large number
of voltage sources, it is useful to use mesh analysis; as this analysis requires that all the sources
in a circuit be voltage sources. Therefore, if there are any current sources in a circuit they are
to be converted into equivalent voltage sources,if, on the other hand, the network has more
current sources,nodal analysis is more useful.
Mesh analysis is applicable only for planar networks. For non-planar circuits mesh analysis
is not applicable .A circuit is said to be planar, if it can be drawn on a plane surface without
crossovers. A non-planar circuit cannot be drawn on a plane surface without a crossover.
Figure 3.2 (a) is a planar circuit. Figure 3.2 (b) is a non-planar circuit and fig. 3.2 (c) is a
planar circuit which looks like a non-planar circuit. It has already been discussed that a loop is
a closed path. A mesh is defined as a loop which does not contain any other loops within it.
To apply mesh analysis, our first step is to check whether the circuit is planar or not and the
second is to select mesh currents. Finally, writing Kirchhoff‘s voltage law equations in terms
of unknowns and solving them leads to the final solution.
(a) (b) (c)
Figure 3.2
14. a b c
R2
R4
± I1
f e d
Observation of the Fig.3.2 indicates that there are two loops abefa,and bcdeb in the network
.Let us assume loop currents I1 and I2with directions as indicated in the figure.
Considering the loop abefa alone, we observe that current I1 is passing through R1, and (I1-I2) is
passing through R2. By applying Kirchhoff’s voltage law, we can write
Vs. =I1R1+R2(I1-I2) (3.1)
R1 R3
Vs
Figure 3.3
Similarly, if we consider the second mesh bcdeb, the current I2 is passing through R3 and R4,
and (I2 – I1) is passing through R2. By applying Kirchhoff’s voltage law around the second mesh,
we have
R2 (I2-I1) + R3I2 +R4I2= 0 (3.2)
By rearranging the above equations, the corresponding mesh current equations are I1
(R1+R2) – I2R2 =Vs.
-I1R2 +(R2+R3+R4) I2=0 (3.3)
By solving the above equations, we can find the currents I1 and I2,.If we observe Fig.3.3, the
circuit consists of five branches and four nodes, including the reference node. The number of
mesh currents is equal to the number of mesh equations
SUPERMESH ANALYSIS :
Suppose any of the branches in the network has a current source, then it is slightly difficult to
apply mesh analysis straight forward because first we should assume an unknown voltage
across the currentsource, writing mesh equation as before, and then relate the source current
to the assigned mesh currents. This is generally a difficult approach. On way to overcome this
difficulty is by applying the supermesh technique. Here we have to choose the kind of
supermesh. A supermesh is constituted by two adjacent loops that have a common current
source. As an example, consider the network shown in the figure 3.9.
I2
15. R4
Here the current source I is in the common boundary for the two meshes 1 and 2. This current
source creates a supermesh, which is nothing but a combination of meshes 1 and 2.
R1I1 + R3(I2-I3)=V Or
R1I1 + R3I2 – R4I3= V
Considering mesh 3, we
have R3(I3-I2)+ R4I3=0
Finally the current I from current source is equal to the difference between two mesh
currents i.e.
I1-I2=I
we have thus formed three mesh equations which we can solve for the three unknown
currents in the network.
R2
+ V I1 I2 R3 I3
-
1 I 2 3
Figure 3.9
16. R2 R4
R1 R3
1 2
R1
NODALANALYSIS:
we discussed simple circuits containing only two nodes, including the reference node. In general, in a N node circuit, one of
the nodes is chosen as the reference or datum node, then it is possible to write N -1nodal equations by assuming N-1 node
voltages. For example,a10 node circuit requires nine unknown voltages and nine equations. Each node in a circuit can be
assigned a number or a letter. The node voltage is the voltage of a given node with respect to one particular node, called the
reference node, which we assume at zero potential. In the circuit shown in fig. 3.12, node 3 is assumed as the Reference node.
The voltage at node 1 is the voltage at that node with respect to node 3. Similarly, the voltage at node 2 is the voltage at that
node with respect to node 3. Applying Kirchhoff’s current law at node 1, the current entering is the current leaving (See
Fig.3.13)
3 2
I1 R5
4 Figure 3.12 R2
I1
Figure 3.13
I1= V1/R1 + (V1-V2)/R2
17. Where V1 and V2 are the voltages at node 1 and 2, respectively. Similarly, at node 2.the
current entering is equal to the current leaving as shown in fig. 3.1
I1= V1/R1 + (V1-V2)/R2 Where V1 and V2 are the voltages at node 1
and 2, respectively. Similarly, at node 2.the current entering is equal to the current leaving
as shown in fig. 3.14
R2 R4
R3 R5
Figure 3.14
(V2-V1)/R2 + V2/R3 + V2/(R4+R5) =0
Rearranging the above equations, we
have V1[1/R1+1/R2]-V2(1/R2)= I1
-V1(1/R2) + V2[1/R2+1/R3+1/(R4+R5)]=0
From the above equations we can find the voltages at each node.
3.3 NODAL EQUATIONS BY INSPECTION METHOD The nodal equations for a general planar network can also be written by
inspection without going through the detailed steps. Consider a three node resistive network, including the reference node, as shown in fig
3.16
R1 R3 R5
V1
V2
In fig. 3.16 the points a and b are the actual nodes and c is the reference node.
Now consider the nodes a and b separately as shown in fig 3.17(a) and (b)
R1 Va R3 R3 Vb R5
Vb Va
V1
a b
R
2
R
4
c
I1 I5 I3
R2
(a)
I3
R4 I4
I5
V2
(b)
18. 1 2 3
R2 VX
R1 R3 R4
VY
In fig 3.17 (a), according to Kirchhoff’s current law we have
I1+I2+I3=0
(Va-V1)/R1 +Va/R2+ (Va-Vb)/R3= 0 (3.39)
In fig 3.17 (b) , if we apply Kirchhoff’s current law
I4+ I5= I3
(Vb-Va)/R3 + Vb/R4+(Vb-V2)/R5=0 (3.40)
Rearranging the above equations we get
(1/R1+1/R2+1/R3)Va-(1/R3)Vb=(1/R1)V1 (3.41)
(-1/R3)Va+ (1/R3+1/R4+1/R5)Vb=V2/R5 (3.42)
In general, the above equation can be written as
GaaVa + GabVb=I1 (3.43)
GbaVa + GbbVb=I2 (3.44)
By comparing Eqs 3.41,3.42 and Eqs 3.43, 3.44 we have the self conductance at node
a, Gaa=(1/R1 + 1/R2 + 1/R3) is the sum of the conductances connected to node a. Similarly, Gbb=
(1/R3 + 1/R4 +1/R5) is the sum of the conductances connected to node b. Gab=(-1/R3) is the
sum of the mutual conductances connected to node a and node b. Here all the mutual
conductances have negative signs. Similarly, Gba= (-1/R3) is also a mutual conductance
connected between nodes b and a. I1 and I2 are the sum of the source currents at node a and
node b, respectively. The current which drives into the node has positive sign, while the
current that drives away from the node has negative sign.
SUPERNODE ANALYSIS
Suppose any of the branches in the network has a voltage source, then it is slightly difficult to
apply nodal analysis. One way to overcome this difficulty is to apply the supernode technique.
In this method, the two adjacent nodes that are connected by a voltage source are reduced to a
single node and then the equations are formed by applying Kirchhoff’s current law as usual. This
is explained with the help of fig. 3.19
V1 V2 + _ V3
I R5
4
FIG 3.19
19. a
It is clear from the fig.3.19, that node 4 is the reference node. Applying Kirchhoff’s current
law at node 1, we get
I=(V1/R1 ) + (V1-V2)/R2
Due to the presence of voltage source Vχ in between nodes 2 and 3 , it is slightly difficult to find out the
current. The supernode technique can be conveniently applied in this case.
Accordingly, we can write the combined equation for nodes 2 and 3 as under.
SOURCE TRANSFORMATION TECHNIQUE
In solving networks to find solutions one may have to deal with energy sources. It has
already been discussed in chapter 1 that basically, energy sources are either voltage sources
or current sources. Sometimes it is necessary to convert a voltage source to a current source
or vice-versa. Any practical voltage source consists of an ideal voltage source in series with an
internal resistance. Similarly, a practical current source consists of an ideal current source in
parallel with an internal resistance as shown in figure3.21. Rv and Ri represent the internal
resistances of the voltage source Vs ,and current source Is ,respectively.
RV
a
VS IS
b fig. 3.21 b
Any source, be it a current source or a voltage source, drives current through its load
resistance, and the magnitude of the current depends on the value of the load resistance. Fig
represents a practical voltage source and a practical current source connected to the same
load resistance RL.
R1
20. NETWORK THEOREMS
Before start the theorem we should know the basic terms of the network.
Circuit: It is the combination of electrical elements through which current
passes is called circuit.
Network: It is the combination of circuits and elements is called network.
Unilateral :It is the circuit whose parameter and characteristics change
with change in the direction of the supply application.
Bilateral: It is the circuit whose parameter and characteristics do not
change with the supply in either side of the network.
Node: It is the inter connection point of two or more than two elements is
called node.
Branch: It is the interconnection point of three or more than three elements
is called branch.
Loop: It is a complete closed path in a circuit and no element or node is
taken more than once.
Super-Position Theorem :
Statement :’’ It states that in a network of linear resistances containing more
than one source the current which flows at any point is the sum of all the
currents which would flow at that point if each source were considered
separately and all other sources replaced for time being leaving its internal
resistances if any’’.
Explanation :
Considering E1 source
21. Step 1.
R2&r are in series and parallel with R3 and again series with R1
(R2+r2) || R3
=
(R2 + r2)R3
= m
R2
+ r2+R3
(say)
Rt1 = m + R1 + r1
I =
E1
5 Rt
I =
1
I1 R3
R2 + r2 + R3
I =
I1 (R2 + r2 )
R2 + r2 +R3
Step – 2
Considering E2 source,R1&r2 are series and R3 parallel and R2 in series
(R1+r1) || R3
=
(R1 + r1)R3
= n
R1
+ r1+R3
(say)
Rt2 = n + R2 + r2
I =
E2
6 Rt
I /
=
I21 (R1 + r1)
7 R + r + R
I /
=
1 1 3
I2 R3
1
R + r + R
2
3
2
22. Thevenin’s Theorem :
The current flowing through the load resistance R1 connected across any two
terminals A and B of a linear active bilateral network is given by
Where Vth = Voc is the open. circuit voltage across A and B terminal when RL is
removed. Ri =Rth is the internal resistances of the network as viewed back into
the open circuit network from terminals A & B with all sources replaced by their
internal resistances if any.
Explanation :
Step – 1 for finding Voc
Remove RL temporarily to find Voc.
I =
E
R1 + R2 + r
Voc = IR2
Step – 2 finding Rth
Remove all the sources leaving their internal resistances if any and viewed from
open circuit side to find out Ri or Rth.
Ri = (R1 + r) || R2
R =
(R1 + r)R2
R1 + r +R2
i
23. Step – 3
Connect internal resistances and Thevenin’s voltage in series with load
resistance RL.
Norton's Theorem
Statement : In any two terminal active network containing voltage sources and
resistances when viewed from its output terminals in equivalent to a constant
current source and a parallel resistance. The constant current source is equal to
the current which would flow in a short circuit placed across the terminals and
parallel resistance is the resistance of the network when viewed from the open
circuit side after replacing their internal resistances and removing all the
sources.
OR
In any two terminal active network the current flowing through the load
resistance RL is given by
I =
I sc Ri
Ri RL
Where Ri is the internal resistance of the network as viewed from the open ckt
side A & B with all sources being replaced by leaving their internal resistances if
any.
Isc is the short ckt current between the two terminals of the load
resistance when it is shorted
Explanation :
Step – 1
L
24. A &B are shorted by a thick copper wire to find out Isc
ISc = E /(R1 + r
Step – 2
Remove all the source leaving its internal resistance if any and viewed from
open circuit side A and B into the network to find Ri .
Ri = (R1 + r) || R2
Ri = (R1 + r)R2 /(R1 + r + R2
Step – 3
Connect Isc & Ri in parallel with RL
I =
I sc Ri
Ri + RL
L
25. L
L
Maximum PowerTransfer Theorem
Statement : A resistive load will abstract maximum power from a network when
the load resistance is equal to the resistance of the network as viewed from the
output terminals(Open circuit) with all sources removed leaving their internal
resistances if any
Proof :
I L = Vth
Ri + RL
Power delivered to the load
resistance is given by
P = I 2
R
V
2
= th
RL
Ri + RL
V
2
R
= th L
(Ri + RL )
Power delivered to the load resistance RL will be maximum
When dPL
= 0
dRL
d V2
R
th L
= 0
dR (R + R )2
L i L
V 2
(R + R )2
− V 2
R 2(R + R )
th i L th L i L
= 0
(Ri + RL )
V 2
(R + R )2
− V 2
R 2(R + R ) = 0
th i L th L i L
V 2
(R + R )2
− 2V 2
R (R + R ) = 0
th i L th L i L
V 2
(R + R )2
= 2V 2
R (R + R )
th i L th L i L
Ri + RL = 2RL
Ri = 2RL −RL
Ri = RL
L
2
4
26. 2
L
V 2
(PL)max =
th
RL
(Ri + RL )
V 2
= th RL
4RL
V 2
= th
RL
4R
2
MILLIMAN’S THEOREM:
According to Millimans Theorem number of sources can be converted into
a single source with a internal resistance connected in series to it,if the sources
are in parallel connection.
According to the Milliman’s theorem the equivalent voltage source
E
1
+ E
1
+ E
1
+ ..
1
R
2
R
3
R
E'= 1 2 3
1
+
1
R1 R2
+
1
+ .....
R3
=
E1G1
+
E2G2
+
E3G3
+ ..
G
1
+
G
2
+
G
3
+
.
.
.
E
1
+
E
2
+
E
3
+
..
=
R1
R2
R3
G1 +
G2 +
G3 + ....
=
I1 + I2 + I3 +..
G1 + G2 + G3 + ...
L
4R
2
V 2
(PL ) max = th
2
27. COMPENSATION THEOREM:
Statement:
It’s states that in a circuit any resistance ‘R” in a branch of network in
which a current ‘I’ is flowing can be replaced. For the purposes of calculations
by a voltage source = - IR
OR
If the resistance of any branch of network is changed from R to R +4R
where the current flowing originaly is i. The change current at any other place
in the network may be calculated by assuming that one e.m.f – I R has been
injected into the modified branch. While all other sources have their e.m.f.
suppressed and ‘R’ represented by their internal resistances only.
Exp – (01)
Calculate the values of new currents in the network illustrated , when the
resistor R3 is increased by 30%.
Solution :- In the given circuit , the values of various branch currents are
I1 = 75/(5 +10) = 5A
I 3 = I 2
=
5 20
40
= 2 .5 Amp .
Now the value of R3, when it increase 30%
R3 = 20 + (20 0.3) = 26
IR = 26 − 20 = 6
V = −IR
= −2.5 6
= −15V
5 || 20 =
5 20
=
100
= 4
5+20 25
I3 '= 15
4 +26
=
15
= 0.5Amp
30
28. I ' =
0.5 5
= 0.1Amp
2
25
I ' =
0.5 20
= 0.4Amp
1
25
p
I2 "= 0.1+ 2.5 = 2.6Amp
I3"= 2.5 − 0.5 = 2 Amp
RECIPROCITY THEOREM :
Statement :
It states that in any bilateral network, if a source of e.m.f ‘E’ in any branch
produces a current ‘I’ any other branch. Then the same e.m.f ‘E’ acting in the
second branch would produce the same current ‘I’ in the 1st
branch.
Step – 1 First ammeter B reads the current in this branch due to the 36v
source, the current is given by
4 || 12 =
4 12
= 3
16
R = 2 + 4 + 3 = 9
I =
36
= 4 Amp
Step – (II) Then interchanging the sources and
measuring the current
6 ||12 =
612
=
72
= 4
6 +12 18
R = 4 + 3 + 1 = 8
I =
36
= 4.5Amp , I = 3Amp Transfer resistance =
V
=
36
=12.
8
A
6+2 I 3
29. Topology:
➢ Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size and
the shape,
➢ Not concerned with the particular types of elements appearing in
the circuit, but only with the way in which branches and nodes
are arranged.
Graph:
➢ A graph corresponding to a given network is obtained by
replacing all circuit elements with lines.
Connected graph: A graph in which at least one path exists
between any two nodes of the graph. If the network has a
transformer as one of the element, then the resulted graph is
unconnected
Directed or Oriented graph: A graph that has all the nodes
and branches numbered and also directions are given to the
branches.
Subgraph: The subset of a graph. If the number of nodes and
branches of a subgraph is less than that of the graph, the
subgraph is said to be proper.
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n 1). This is also the rank of the graph to
which the tree belongs.
Twigs:
- The branches of a tree are known as twigs,
Links or Chords: - The branches that are removed from the graph while forming tree are
termed as links or chords, Links are complement of twigs.