BEE-101 Unit 1: DC Circuits (PDF)

Course Material OF FUNDAMENTALS OF ELECTRICAL ENGINEERING BEE-101/BEE-201 Session 2024-25 Course Coordinator: Mr. Vinod Kumar Course Teachers: Dr. V. S. Gupta Dr. Sunil Kumar Chaudhary Mr. Manish Shrivastava Dr. Bhuvn...

Course Material OF FUNDAMENTALS OF ELECTRICAL ENGINEERING BEE-101/BEE-201 Session 2024-25 Course Coordinator: Mr. Vinod Kumar Course Teachers: Dr. V. S. Gupta Dr. Sunil Kumar Chaudhary Mr. Manish Shrivastava Dr. Bhuvnesh Khokhar Mr. Ravindra Kumar Yadav Dr. Nitesh Singh Dr. Pawan Kumar Kushwaha Mr. Chandan Kumar Mr. Kumar Prabhakar Dr. Smruti Ranjan Jagadeb BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  Unit 1: DC Circuits CO1: Apply Kirchhoff’s laws and other methods in solving DC circuits Contents: Electrical circuit elements (R, L and C), Concept of active and passive elements, voltage and current sources, concept of linearity and linear network, unilateral and bilateral elements, Kirchhoff‟s laws, Loop and nodal methods of analysis. 1. Electrical circuit elements (R, L and C): The interconnection of various electric elements in a prescribed manner comprises as an electric circuit in order to perform a desired function. The electric elements include controlled and uncontrolled source of energy, resistors, capacitors, inductors, etc. Analysis of electric circuits refers to computations required to determine the unknown quantities such as voltage, current and power associated with one or more elements in the circuit. To contribute to the solution of engineering problems one must acquire the basic knowledge of electric circuit analysis and laws. We shall discuss briefly some of the basic circuit elements and the laws that will help us to develop the background of subject. a) Resistor: Resistor is a dissipative element, which converts electrical energy into heat when the current flows through it in any direction. The law governing the current into and voltage across a resistor is: (i) The relationship is known as Ohm’s law. But resistor can be regarded as linear only within the specified limits, outside which the behavior becomes non-linear. The resistance of a resistor is temperature dependent and rises with temperature. Mathematically it can be represented as: (ii) Where = Resistance at and = Resistance at Temperature coefficient and it may be positive and negative both Temperature in And power dissipated by resistor is Watts Resistor is represented by the symbol Unit of Resistance is ohm (𝛀) b) Capacitor (C): It is a two terminal element that has the capability of energy storage in electric field. The law governing the relationship of capacitor is: (iii) After integrating equation (iii), we get BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  ∫ (iv) Where Capacitor voltage at , for initially uncharged capacitor Hence, ∫ (v) The above expressions show that the voltage of a capacitor cannot change instantaneously. Energy stored in capacitor can be represented by ∫ ∫ ∫ (ix) Capacitor is represented by the symbol Unit of Capacitance is Farad (F) c) Inductor (L): It is a two-terminal storage element in which energy is stored in the magnetic field. The relation of an inductance is: (vi) After integrating expression (vi), we get ∫ (vii) Where Inductor current at , for initially if current through inductor Hence, ∫ (viii) The above expressions show that the current through an inductor cannot change instantaneously. Energy stored in inductor can be represented by ∫ ∫ ∫ (ix) Inductor is represented by the symbol Unit of Inductance is Henry (H) 2. Concept of active and passive elements: Electrical Network: Any possible combination of various electric elements (Resistor, Inductor, Capacitor, Voltage source, Current source) connected in any manner what so ever is called an electrical network. We may classify circuit elements in two categories, passive and active elements. BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  Electrical Circuit: An electric circuit is a closed energized electric network. It means circuit must have closed path with energy sources. From the above example, we can say that fig 1 and fig 2 are electric networks but only fig 2 is electric circuit. It means, electric circuit is always an electric network but electric network may or may not be an electric circuit. Passive Element: The element which receives energy (or absorbs energy) and then either converts it into heat (R) or stored it in an electric (C) or magnetic (L ) field is called passive element, and the network containing these elements without energy sources are known as passive network. Examples are resistor, inductor, capacitor, transformer etc. Active Element: The elements that supply energy to the circuit is called active element and the network containing these sources together with other circuit elements are known as active network. Examples of active elements include voltage and current sources, generators, and electronic devices that require power supplies. A transistor is an active circuit element, meaning that it can amplify power of a signal. 3. Energy Sources (Voltage and Current Sources): There are two types of energy sources namely Voltage Sources and Current Sources. Energy Sources Voltage Source Current Source Independent Dependent Independent Dependent Voltage Source Voltage Source Current Source Current Source Voltage Dependent Current Dependent Voltage Dependent Current Dependent Voltage Source Voltage Source Current Source Current Source (VDVS) (CDVS) (VDCS) (CDCS) BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida Here, we shall study only about independent voltage source and independent current source. a) Independent Voltage Source: A hypothetical generator which maintains its value of voltage independent of the output current. It can be represented as: Fig: Ideal DC Voltage Source Fig: Practical DC Voltage Source If the value of internal resistance will be zero, then the voltage source is called as ideal voltage source. The V-I characteristics for ideal and practical voltage source is given below: Fig: V-I Characteristic of Voltage Source b) Independent Current Source: A generator which maintains its output current independent of the voltage across its terminals. It can be represented as: Fig: Ideal DC Current Source Fig: Practical DC Current Source if the value of internal resistance will be infinity, then the current source is called as ideal current source. The V-I characteristics for ideal and practical current source is given below: BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  Fig: V-I Characteristic of Current Source 4. Concept of Linearity and Linear Network: For a network to be linear, it should have to follow the principle of superposition and homogeneity both. Principle of Superposition: An element or circuit obeys the principle of superposition if the net effect of the sum of causes equals the sum of their individual effects. Mathematically, let cause x and effect y be related as: (i) Let the cause be scaled by a factor. Then the functional relationship obeys homogeneity, if (ii) Consider two causes , then Let the combined effect of these two causes be scaled by and respectively. The principle of superposition then yields if: (iii) If homogeneity is also satisfied, then (iv) A functional relationship is said to be linear if it obeys both superposition and homogeneity. Any element governed by such a functional relationship is linear. A circuit composed of such elements would also be linear. 5. Unilateral and Bilateral Elements: Bilateral Elements: If by reversing the terminal connections of an element in a circuit, the circuit response remains same. Such elements are known as bilateral elements. Examples are Resistor, Inductor, Capacitor etc. Unilateral Elements: If by reversing the terminal connections of an element in a circuit, the circuit response gets change. Such elements are called as unilateral elements. Examples are Voltage Source, Current Source, Diode etc. BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida 6. Kirchhoff‟s laws: There are two types of Kirchhoff’s Law. 1. Kirchhoff’s First Law or Kirchhoff’s Current Law (KCL) 2. Kirchhoff’s Second Law or Kirchhoff’s Voltage Law (KVL) 1. Kirchhoff’s First Law or Kirchhoff’s Current Law (KCL): Kirchhoff’s current law states that, in a given electric circuit, algebraic sum of all the currents meeting at a junction is always zero. In another way we can say that, the total current flowing towards a junction is equal to the total current flowing away from that junction. This law works on the principle of conservation of charge. Sign Convention: If we take direction of current towards the junction as positive (+) sign then direction of current away from the junction will be taken as negative (-)sign or vice-versa. According to Kirchhoff’s Current Law in the above circuit diagram: ∑ 2. Kirchhoff’s Second Law or Kirchhoff’s Voltage Law (KVL): Kirchhoff’s voltage law states that, “ In any electric circuit, the algebraic sum of the voltage drops across the circuit elements of any closed path (or loop or mesh) is equal to the algebraic sum of the EMFs in the path”. BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  In other words, “ The algebraic sum of all the branch voltages around any closed path or closed loop is always zero”. This law works on the principle of conservation of energy. Limitation of this law is that it can only be applied to planner network. Sign Convention: If we take voltage rise with positive (+) sign then voltage drop will be taken with negative (-) sign or vice-versa. When current direction will be from negative terminal to positive terminal, voltage will rise and vice-versa. In all the passive elements current entering terminal is taken as positive and current leaving terminal is taken as negative. According to KVL in the above circuit diagram: 7. Loop and Nodal Methods of Analysis: For study of Loop and Nodal Methods of Analysis, knowledge of basic fundamentals are essential. Some Basic Definitions: 1. Node: A node of a network is an equipotential surface at which two or more circuit elements are joined. 2. Junction: A junction is that point in an electric circuit where three or more elements are joined. So, we can say that junction is always a node but node may or may not be a junction. 3. Loop: A loop is any closed path of the electric network. 4. Mesh: A mesh is the most elementary form of loop, and it cannot be further subdivided into other loops. So, we can say that mesh is always a loop but loop may or may not be a mesh. BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  5. Lumped Network: A network in which physically separate resistors, capacitors and inductors can be represented. 6. Distributed Network: One in which resistors, capacitors, and inductors cannot be physically separated and individually isolated as separate elements. For example, Transmission Line. Loop or Mesh Analysis Method: Mesh analysis is also known as loop analysis method. Mesh analysis is used to find the currents and voltages in a particular circuit. Suppose in a particular electrical circuit , total number of branches are b, total number of nodes are n and total number of junctions are j, then total number of meshes ‘m’ can be calculated by using the following expression: (i) Also, (ii) Note: One thing make sure, when we consider nodes in the given electric circuit then branches will be counted according to the number of nodes, and equation (i) is used to calculate the total number of meshes. When we consider junction in that electrical circuit then branches will be counted according to junctions, and equation (ii) is used to calculate the total number of meshes. Using both the methods same number of meshes will be found for a particular circuit. The independent mesh equations can be obtained by applying KVL to each independent mesh. Mesh current is that current which flows around the perimeter of a mesh. Mesh currents may or may not have a direct identification with branch currents. Mesh currents on the other hand, are fictitious quantity which is introduced because they allow us to solve problems in terms of a minimum number of unknowns. Procedure of Mesh Analysis Method: Step 1: Draw the circuit in which mesh currents or branch currents have to find. Calculate number of independent mesh equations by using the formula given below: (i) Also, (ii) Step 2: Assume independent mesh currents for each mesh. You can choose any direction of the mesh current, but once you have chosen the direction of current, it should remain same throughout the question. BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  Step 3: Apply KVL for each mesh and write the expressions in terms of unknown mesh currents. Step 4: Now, solve the equations by any method, either by simultaneous equation method or Cramer’s Rule and find the unknown values. Example 1a: In the given circuit, find different Branch currents by using mesh analysis method. Solution: Step 1: Calculate total no. of meshes in the given circuit: Here M=2 Step 2: Assume mesh currents in different Meshes. Let the mesh currents are and direction of mesh currents are taken as clockwise. Step 3: Apply KVL for mesh (1) (1) Apply KVL for mesh (2) (2) Step 4: After solving equation (1) and (2) mesh currents can be calculated. Procedure for solving the question using mesh analysis method if the circuit contains current sources There are two cases arise having current sources in the circuit. Case 1: Current source is present outer arm of the mesh Case 2: Current Source is present in between the meshes BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  Case 1: Current source is present outer in arm of the mesh Step 1: Calculate total no. of meshes in the circuit. Here Total no. of meshes = 2 Step 2: Assume mesh currents in different meshes and their direction in those meshes. Step 3: Apply KVL to all the meshes where no current sources are present. But we need not to apply KVL in that mesh where current source is connected in its outer arm of the mesh. The value of mesh current in that mesh will equal to the value of current source. Step 4: Solve the equations and find the mesh currents. Apply KVL in mesh (1) (1) In mesh (2) (2) Case 2: Current Source is present in between the meshes. This case is called the concept of super mesh. Step 1 and Step 2 is same but in Step 3 We create a super mesh by excluding the current source and any elements connected in series with it while taking KVL. Note: A super mesh requires the application of both KVL and KCL. Let us solve such circuit as shown below. Converting Fig. having two meshes into a circuit having single super mesh BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida  Apply KVL in that super mesh (1) Apply KCL at the junction where current source is connected. (2) After solving equation (1) and (2), we can get the value of mesh currents in the circuit. Nodal Analysis Method: 1. Node: A node of a network is an equipotential surface at which two or more circuit elements are joined. 2. Junction: A junction is that point in an electric circuit where three or more elements are joined. So, we can say that junction is always a node but node may or may not be a junction. Circuit analysis by this method is done by using the KCL at the junction of the given circuit. Suppose, total numbers of junctions are j in a particular electrical circuit, then total number of node equations N can be calculated by using the following formula: Total number of Node Equations (i) BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida Procedure of Nodal Methods of Analysis for Case 1: Step1: Identify all nodes (Junctions) in the circuit. Select one node (Junction) as the reference node (assign as ground potential or zero potential) and label the remaining nodes (Junctions) as unknown node (Junction) voltages with respect to the reference node (Junction). Calculate the total number of Node equations using the formula: (i) Step 2: Assign branch currents in each branch. (The choice of direction is arbitrary). Step 3: Apply KCL for each junction except reference and express the branch currents in terms of node assigned voltages. Step 4: Finally solve the equations using different methods. NOTE: It is convenient to select the reference node at the bottom of a circuit or the node that has the maximum number of branches connected to it. Example 1: BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida Procedure of Nodal Methods of Analysis for Case 2 : Step1: Identify all nodes (Junctions) in the circuit. Select one node (Junction) as the reference node (assign as ground potential or zero potential) and label the remaining nodes (Junctions) as unknown node (Junction) voltages with respect to the reference node (Junction). Calculate the total number of Node equations using the formula: (i) Step 2: Assign branch currents in each branch. (The choice of direction is arbitrary). Step 3: Apply KCL for each junction except reference and express the branch currents in terms of node assigned voltages. Step 4: Apply KVL in the mesh containing super node branch. Step 5: Finally solve the equations using different methods. NOTE: It is convenient to select the reference node at the bottom of a circuit or the node that has the maximum number of branches connected to it. Example 2: BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida There are four unknowns and four equations, so we can find out the node voltages and branch currents of the given circuit. Case 3: If super node forms between reference node and non-reference node. BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida BEE-101/BEE-201 Fundamentals of Electrical Engineering GCET, Greater Noida