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Transcript

Basic Electrical Engineering Class notes Introduction 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. Many other systems, like mechanical, hydraulic, thermal, magnetic and power system are easy to analyze and model by a circuit. To learn how to analyze the models of these systems, first one needs to learn the techniques of circuit analysis. We shall discuss briefly some of the basic circuit elements and the laws that will help us to develop the background of subject. Charge: In an object comprised of many atoms, the net charge is equal to the arithmetic sum, taking polarity into account, of the charges of all the atoms taken together. In a massive sample, this can amount to a considerable quantity of elementary charges. The unit of electrical charge in the International System of Units is the coulomb (symbolized C), where 1 C is equal to approximately 6.24 x 1018 elementary charges. It is not unusual for real-world objects to hold charges of many coulombs. An electric field, also called an electrical field or an electrostatic field, surrounds any object that has charge. The electric field strength at any given distance from an object is directly proportional to the amount of charge on the object. Near any object having a fixed electric charge, the electric field strength diminishes in proportion to the square of the distance from the object (that is, it obeys the inverse square law). When two objects having electric charge are brought into each other's vicinity, an electrostatic force is manifested between them. (This force is not to be confused with electromotive force, also known as voltage.) If the electric charges are of the same polarity, the electrostatic force is repulsive. If the electric charges are of opposite polarity, the electrostatic force is attractive. In free space (a vacuum), if the charges on the two nearby objects in coulombs are q1 and q2 and the centers of the objects are separated by a distance r in meters, the net force F between the objects, in newtons, is given by the following formula: F = (q1q2) / (4 o r2) Where o is the permittivity of free space, a physical constant, and is the ratio of a circle's circumference to its diameter, a dimensionless mathematical constant. A positive net force is repulsive, and a negative net force is attractive. This relation is known as Coulomb's law. Electric Current: An electric current is a flow of electric charge. In electric circuits this charge is often carried by moving electrons in a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in a plasma. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 1 Basic Electrical Engineering Class notes The SI unit for measuring an electric current is the ampere, which is the flow of electric charge across a surface at the rate of one coulomb per second. Electric current is measured using a device called an ammeter.[2] Electric currents can have many effects, notably heating, but they also create magnetic fields, which are used in motors, inductors and generators. Electric power: Electric power, like mechanical power, is the rate of doing work, measured in watts, and represented by the letter P. The term wattage is used colloquially to mean "electric power in watts." The electric power in watts produced by an electric current I consisting of a charge of Q coulombs every t seconds passing through an electric potential (voltage) difference of V is Where Q is electric charge in coulombs t is time in seconds I is electric current in amperes V is electric potential or voltage in volts Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. Any electrical network can be analyzed as multiple, interconnected electrical elements in a schematic diagram or circuit diagram, each of which affects the voltage in the network or current through the network. These ideal electrical elements represent real, physical electrical or electronic components but they do not exist physically and they are assumed to have ideal properties according to a lumped element model, while components are objects with less than ideal properties, a degree of uncertainty in their values and some degree of nonlinearity, each of which may require a combination of multiple electrical elements in order to approximate its function. Fig 1.1: Different types of circuit element Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 2 Basic Electrical Engineering Class notes Three passive elements: Resistance , measured in ohms – produces a voltage proportional to the current flowing through the element. Relates voltage and current according to the relation . Capacitance , measured in farads – produces a current proportional to the rate of change of voltage across the element. Relates charge and voltage according to the relation . Inductance , measured in henries – produces the magnetic flux proportional to the rate of change of current through the element. Relates flux and current according to the relation . Basic Elements & Introductory Concepts Electrical Network: A 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. 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. Active Element: The elements that supply energy to the circuit is called active element. 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. On the other hand, transformer is not an active element because it does not amplify the power level and power remains same both in primary and secondary sides. Transformer is an example of passive element. Bilateral Element: Conduction of current in both directions in an element (example: Resistance; Inductance; Capacitance) with same magnitude is termed as bilateral element. Fig 1.2: Sign convention of current in the resistive circuit Unilateral Element: Conduction of current in one direction is termed as unilateral (example: Diode, Transistor) element. Fig 1.3: Biasing of a diode Meaning of Response: An application of input signal to the system will produce an output signal, the behavior of output signal with time is known as the response of the system. Linear and Nonlinear Circuits Linear Circuit: Roughly speaking, a linear circuit is one whose parameters do not change with voltage or current. More specifically, a linear system is one that satisfies (i) homogeneity property [response of ()utα equals α times the response of , ()ut(()Sut α = (())Sutα for all α; and ] (ii) additive Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 3 Basic Electrical Engineering Class notes property [that is the response of system due to an input (()ut1122()()ututαα+) equals the sum of the response of input 11()utα and the response of input22()utα, 1122(()(Sutut αα+ = 1122(())(())SutSutαα+.] When an input is applied to a system “”, the corresponding output response of the system is observed as respectively. Fig. 3.1 explains the meaning of homogeneity and additive properties of a system. Fig 1.4: Block diagram of linear circuit Non-Linear Circuit: Roughly speaking, a non-linear system is that whose parameters change with voltage or current. More specifically, non-linear circuit does not obey the homogeneity and additive properties. Volt-ampere characteristics of linear and non-linear elements are shown in figs. 3.2 - 3.3. In fact, a circuit is linear if and only if its input and output can be related by a straight line passing through the origin as shown in fig.3.2. Otherwise, it is a nonlinear system. Fig 1.5: V-I characteristics of linear circuit Fig 1.6: V-I characteristics of non-linear circuit Potential Energy Difference: The voltage or potential energy difference between two points in an electric circuit is the amount of energy required to move a unit charge between the two points. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 4 Basic Electrical Engineering Class notes Open Circuit and Short Circuit Open Circuit: An electric circuit that has been broken, so that there is no complete path for current flow. A condition in an electric circuit in which there is no path for current between two points; examples are a broken wire and a switch in the open, or off, position. Open-circuit voltage is the potential difference between two points in a circuit when a branch (current path) between the points is open-circuited. Open-circuit voltage is measured by a voltmeter which has a very high resistance (theoretically infinite). Short Circuit : A low-resistance connection established by accident or intention between two points in an electric circuit. The current tends to flow through the area of low resistance, bypassing the rest of the circuit. Common usage of the term implies an undesirable condition arising from failure of electrical insulation, from natural causes (lightning, wind, and so forth), or from human causes (accidents, intrusion, and so forth). In circuit theory the short-circuit condition represents a basic condition that is used analytically to derive important information concerning the network behavior and operating capability. Thus, along with the open-circuit voltage, the short-circuit current provides important basic information about the network at a given point. The short-circuit condition is also used in network theory to describe a general condition of zero voltage. Resistivity and Conductivity: Resistivity is the electrical resistance offered by a homogeneous unit cube of material to the flow of a direct current of uniform density between opposite faces of the cube. Also called specific resistance, it is an intrinsic, bulk (not thin-film) property of a material. Resistivity is usually determined by calculation from the measurement of electrical resistance of samples having a known length and uniform cross section according to the following equation where ? is the resistivity, R the measured resistance, A the cross-sectional area, and l the length. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 5 Basic Electrical Engineering Class notes In the mks system (SI), the unit of resistivity is the ohm-meter. Therefore, in the equation below, resistance is expressed in ohms, and the sample dimensions in meters. Resistivity is also temperature dependent. Resistivity increases by about 0.4%/K at room temperature and is nearly proportional to the absolute temperature over wide temperature ranges. The roomtemperature resistivity of pure metals extends from approximately 1.5 × 10^-8 ohm-meter for silver, the best conductor, to 135 × 10^-8 ohm-meter for manganese, the poorest pure metallic conductor. Most metallic alloys also fall within the same range. Insulators have resistivity within the approximate range of 10^8 to 10^16 ohm-meters. The resistivity of semiconductor materials, such as silicon and germanium, depends not only on the basic material but to a considerable extent on the type and amount of impurities in the base material. Large variations result from small changes in composition, particularly at very low concentrations of impurities. Values typically range from 10^-4 to 10^5 ohmmeters. Conductivity: Electrical conductivity is a measure of a material's ability to conduct an electric current. When an electric potential difference is placed across a conductor, its movable charges flow, giving rise to an electric current. The conductivity σ is defined as the ratio of the current density J to the electric field strength E: Conductivity is the reciprocal (inverse) of electrical resistivity and has the SI units of Siemens per metre (S·m-1) i.e. if the electrical conductance between opposite faces of a 1-metre cube of material is 1 Siemens then the material's electrical conductivity is 1 Siemens per metre. Electrical conductivity is commonly represented by the Greek letter σ, but κ or γ are also occasionally used. Classification of materials by conductivity: A conductor such as a metal has high conductivity. An insulator like glass or a vacuum has low conductivity. The conductivity of a semiconductor is generally intermediate, but varies widely under different conditions, such as exposure of the material to electric fields or specific frequencies of light, and, most important, with temperature. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 6 Basic Electrical Engineering Class notes Temperature Dependence: Electrical conductivity is more or less strongly dependent on temperature. In metals, electrical conductivity decreases with increasing temperature, whereas in semiconductors, electrical conductivity increases with increasing temperature. Over a limited temperature range, the electrical conductivity can be approximated as being directly proportional to temperature. In order to compare electrical conductivity measurements at different temperatures, they need to be standardized to a common temperature. This dependence is often expressed as a slope in the conductivity-vs-temperature graph, and can be used: Where σT′ is the electrical conductivity at a common temperature, T′ σT is the electrical conductivity at a measured temperature, T α is the temperature compensation slope of the material, T is the measured temperature, T′ is the common temperature. The temperature compensation slope for most naturally occurring waters is about 2 %/°C, however it can range between (1 to 3) %/°C. This slope is influenced by the geochemistry, and can be easily determined in a laboratory. Series and Parallel Circuits Series and parallel electrical circuits are two basic ways of wiring components. The names describe the method of attaching components, which is one after the other or next to each other. It is said that two circuit elements are connected in parallel if the ends of one circuit element are connected directly to the corresponding ends of the other. If the circuit elements are connected end to end, it is said that they are connected in series. A series circuit is one that has a single path for current flow through all of its elements. A parallel circuit is one that requires more than one path for current flow in order to reach all of the circuit elements. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 7 Basic Electrical Engineering Class notes With each of the two basic circuit (series and parallel) configurations, we have specific sets of rules describing voltage, current, and resistance relationships. Series circuits: Voltage drops add to equal total voltage. All components share the same (equal) current. Resistances add to equal total resistance. Parallel circuits: All components share the same (equal) voltage. Branch currents add to equal total current. Resistances diminish to equal total resistance. Resistances in Series: To find the total resistance of all the components, add the individual resistances of each component: Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 8 Basic Electrical Engineering Class notes for components in series with resistances R1, R2, etc. To find the current I use Ohm's law: To find the voltage across a component with resistance Ri, use Ohm's law again: Vi = I×Ri Where I is the current, as calculated above. The components divide the voltage according to their resistances, so, in the case of two resistors, V1/V2=R1/R2. Resistances in Parallel: Voltages across components in parallel with each other are the same in magnitude and they also have identical polarities. Hence, the same voltage variable is used for all circuits’ elements in such a circuit. The total current I is the sum of the currents through the individual loops, found by Ohm's Law. Factoring out the voltage gives Notation The parallel property can be represented in equations by two vertical lines (as in geometry) to simplify equations. For two resistors, Resistors To find the total resistance of all components, add the reciprocals of the resistances R i of each component and take the reciprocal of the sum: Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 9 Basic Electrical Engineering Class notes To find the current in a component with resistance Ri, use Ohm's law again: The components divide the current according to their reciprocal resistances, so, in the case of two resistors, I1/I2=R2/R1 Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 10 Basic Electrical Engineering Class notes Characteristics of circuit element Electrical Resistance The electrical resistance of an electrical conductor is the opposition to the passage of an electric current through that conductor. The inverse quantity is electrical conductance, the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with the notion of mechanical friction. The SI unit of electrical resistance is the ohm (Ω), while electrical conductance is measured in Siemens (S). An object of uniform cross section has a resistance proportional to its resistivity and length and inversely proportional to its cross-sectional area. All materials show some resistance, except for superconductors, which have a resistance of zero. The resistance (R) of an object is defined as the ratio of voltage across it (V) to current through it (I), while the conductance (G) is the inverse: For a wide variety of materials and conditions, V and I are directly proportional to each other, and therefore R and G are constant (although they can depend on other factors like temperature or strain). This proportionality is called Ohm's law, and materials that satisfy it are called "Ohmic" materials. In other cases, such as a diode or battery, V and I are not directly proportional, or in other words the I–V curve is not a straight line through the origin, and Ohm's law does not hold. In this case, resistance and conductance are less useful concepts, and more difficult to define. The ratio V/I is sometimes still useful, and is referred to as a "chordal resistance" or "static resistance", as it corresponds to the inverse slope of a chord between the origin and an I–V curve. In other situations, the derivative may be most useful; this is called the "differential resistance". Conductors and Resistor Substances in which electricity can flow are called conductors. A piece of conducting material of a particular resistance meant for use in a circuit is called a resistor. Conductors are made of highconductivity materials such as metals, in particular copper and aluminum. Resistors, on the other hand, are made of a wide variety of materials depending on factors such as the desired resistance, amount of energy that it needs to dissipate, precision, and costs. Relation to Resistivity and Conductivity The resistance of a given object depends primarily on two factors: What material it is made of, and its shape. For a given material, the resistance is inversely proportional to the cross-sectional area; for example, a thick copper wire has lower resistance than an otherwise-identical thin copper wire. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 11 Basic Electrical Engineering Class notes Also, for a given material, the resistance is proportional to the length; for example, a long copper wire has higher resistance than an otherwise-identical short copper wire. The resistance Rand conductance G of a conductor of uniform cross section, therefore, can be computed as where is the length of the conductor, measured in metres [m], A is the cross-sectional area of the conductor measured in square [m²], σ (sigma) is the electrical conductivity measured in siemens per meter (S·m−1), and ρ (rho) is the electrical (also called specific electrical resistance) of the material, measured in ohm-meters (Ω·m). The resistivity and conductivity are proportionality constants, and therefore depend only on the material the wire is made of, not the geometry of the wire. Resistivity and conductivity are reciprocals: ability to oppose electric current. . Resistivity is a measure of the material's This formula is not exact, as it assumes the current density is totally uniform in the conductor, which is not always true in practical situations. However, this formula still provides a good approximation for long thin conductors such as wires. Another situation for which this formula is not exact is with alternating current (AC), because the skin effect inhibits current flow near the center of the conductor. For this reason, the geometrical cross-section is different from the effective cross-section in which current actually flows, so resistance is higher than expected. Similarly, if two conductors near each other carry AC current, their resistances increase due to the proximity effect. At commercial power frequency, these effects are significant for large conductors carrying large currents, such as busbars in an electrical substation,[3] or large power cables carrying more than a few hundred amperes. What determines resistivity? The resistivity of different materials varies by an enormous amount: For example, the conductivity of teflon is about 1030times lower than the conductivity of copper. Why is there such a difference? Loosely speaking, a metal has large numbers of "delocalized" electrons that are not stuck in any one place, but free to move across large distances, whereas in an insulator (like teflon), each electron is tightly bound to a single molecule, and a great force is required to pull it away. Semiconductors lie between these two extremes. More details can be found in the article: Electrical resistivity and conductivity. For the case of electrolyte solutions, see the article: Conductivity (electrolytic). Resistivity varies with temperature. In semiconductors, resistivity also changes when exposed to light. See. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 12 Basic Electrical Engineering Class notes Measuring Resistance An instrument for measuring resistance is called an ohmmeter. Simple ohmmeters cannot measure low resistances accurately because the resistance of their measuring leads causes a voltage drop that interferes with the measurement, so more accurate devices use four-terminal sensing. Inductor An inductor, also called a coil or reactor, is a passive two-terminal electrical component which resists changes in electric current passing through it. It consists of a conductor such as a wire, usually wound into a coil. When a current flows through it, energy is stored temporarily in a magnetic field in the coil. When the current flowing through an inductor changes, the time-varying magnetic field induces a voltage in the conductor, according to Faraday’s law of electromagnetic induction, which opposes the change in current that created it. Fig 2.1: Different types of inductor Capacitor A capacitor (originally known as a condenser) is a passive two-terminal electrical component used to store energy electrostatically in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors (plates) separated by a dielectric (i.e. insulator). The conductors can be thin films, foils or sintered beads of metal or conductive electrolyte, etc. The "nonconducting" dielectric acts to increase the capacitor's charge capacity. A dielectric can be glass, ceramic, plastic film, air, vacuums, paper, mica, oxide layer etc. Capacitors are widely used as parts of electrical circuits in many common electrical devices. Unlike a resistor, an ideal capacitor does not dissipate energy. Instead, a capacitor stores energy in the form of an electrostatic field between its plates. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 13 Basic Electrical Engineering Class notes When there is a potential difference across the conductors (e.g., when a capacitor is attached across a battery), an electric field develops across the dielectric, causing positive charge +Q to collect on one plate and negative charge −Q to collect on the other plate. If a battery has been attached to a capacitor for a sufficient amount of time, no current can flow through the capacitor. However, if a time-varying voltage is applied across the leads of the capacitor, a displacement current can flow. An ideal capacitor is characterized by a single constant value for its capacitance. Capacitance is expressed as the ratio of the electric charge Q on each conductor to the potential difference V between them. The SI unit of capacitance is the farad (F), which is equal to one coulomb per volt (1 C/V). Typical capacitance values range from about 1 pF (10−12 F) to about 1 mF (10−3 F). The capacitance is greater when there is a narrower separation between conductors and when the conductors have a larger surface area. In practice, the dielectric between the plates passes a small amount of leakage current and also has an electric field strength limit, known as the breakdown voltage. The conductors and leads introduce an undesired inductance and resistance. Capacitors are widely used in electronic circuits for blocking direct current while allowing alternating current to pass. In analog filter networks, they smooth the output of power supplies. In resonant circuits they tune radios to particular frequencies. In electric power transmission systems, they stabilize voltage and power flow. Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship: Where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 14 Basic Electrical Engineering Class notes Sign Convention In electrical engineering, the passive sign convention (PSC) is a sign convention or arbitrary standard rule adopted universally by the electrical engineering community for defining the sign of electric power in an electric circuit. The convention defines electric power flowing out of the circuit into an electrical component as positive, and power flowing into the circuit out of a component as negative. So a passive component which consumes power, such as an appliance or light bulb, will have positive power dissipation, while an active component, a source of power such as an electric generator or battery, will have negative power dissipation. This is the standard definition of power in electric circuits. To comply with the convention, the direction of the voltage and current variables used to calculate power and resistance in the component must have a certain relationship: the current variable must be defined so positive current enters the positive voltage terminal of the device. These directions may be different from the directions of the actual current flow and voltage. Reference direction The power flow p and resistance r of an electrical component are related to the voltage v and current i variables by the defining equation for power and Ohm's law: Like power, voltage and current are signed quantities. The current flow in a wire has two possible directions, so when defining a current variable i the direction which represents positive current flow must be indicated, usually by an arrow on the circuit diagram. This is called the reference direction for current i. If the actual current is in the opposite direction, the variable i will have a negative value. Similarly in defining a voltage variable v, the terminal which represents the positive side must be specified, usually with an arrow or plus sign. This is called the reference direction for voltagev. To understand the PSC, it is important to distinguish the reference directions of the variables, v and i, which can be assigned at will, from the direction of the actual voltage and current, which is determined by the circuit. The idea of the PSC is that by assigning the reference direction of variables v and i in a component with the right relationship, the power flow in passive components calculated from Eq. (1) will come out positive, while the power flow in active components will come out negative. It is not necessary to know whether a component produces or consumes power when analyzing the circuit; reference directions can be assigned arbitrarily, directions to currents and polarities to voltages, then the PSC is used to calculate the power in components. If the power comes out positive, the component is a load, converting electric power to some other kind of power. If the power comes out negative, the component is a source, converting some other form of power to electric power. Sign conventions The above discussion shows that choosing the relative direction of the voltage and current variables in a component determines the direction of power flow that is considered positive. The reference directions of the individual variables are not important, only their relation to each other. There are two choices: Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 15 Basic Electrical Engineering Class notes Passive sign convention: Defining the current variable as entering the positive terminal means that if the voltage and current variables have positive values, current flows from the positive to the negative terminal, doing work on the component, as occurs in a passive component. So power flowing into the component from the line is defined as positive; the power variable represents power dissipation in the component. Therefore Active components (power sources) will have negative resistance and negative power flow Passive components (loads) will have positive resistance and positive power flow This is the convention normally used. Active sign convention: Defining the current variable as entering the negative terminal means that if the voltage and current variables have positive values, current flows from the negative to the positive terminal, so work is being done on the current, and power flows out of the component. So power flowing out of the component is defined as positive; the power variable represents power produced. Therefore: Active components will have positive resistance and positive power flow Passive components will have negative resistance and negative power flow This convention is rarely used, except for special cases in power engineering. In practice it is not necessary to assign the voltage and current variables in a circuit to comply with the PSC. Components in which the variables have a "backward" relationship, in which the current variable enters the negative terminal, can still be made to comply with the PSC by changing the sign of the constitutive relations (1) and (2) used with them. A current entering the negative terminal is equivalent to a negative current entering the positive terminal, so in such a component , and Kirchhoff’s Laws Kirchhoff’s laws are basic analytical tools in order to obtain the solutions of currents and voltages for any electric circuit; whether it is supplied from a direct-current system or an alternating current system. But with complex circuits the equations connecting the currents and voltages may become so numerous that much tedious algebraic work is involve in their solutions. Elements that generally encounter in an electric circuit can be interconnected in various possible ways. Before Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 16 Basic Electrical Engineering Class notes discussing the basic analytical tools that determine the currents and voltages at different parts of the circuit, some basic definition of the following terms are considered. Fig 3.1: A simple resistive circuit Node- A node in an electric circuit is a point where two or more components are connected together. This point is usually marked with dark circle or dot. The circuit in fig. 3.4 has nodes a, b, c, and g. Generally, a point, or a node in a circuit specifies a certain voltage level with respect to a reference point or node. Branch- A branch is a conducting path between two nodes in a circuit containing the electric elements. These elements could be sources, resistances, or other elements. Fig.3.4 shows that the circuit has six branches: three resistive branches (a-c, b-c, and b-g) and three branches containing voltage and current sources (a-, a-, and c-g). Loop- A loop is any closed path in an electric circuit i.e., a closed path or loop in a circuit is a contiguous sequence of branches which starting and end points for tracing the path are, in effect, the same node and touches no other node more than once. Fig. 3.4 shows three loops or closed paths namely, a-b-g-a; b-c-g-b; and a-c-b-a. Further, it may be noted that the outside closed paths a-c-g-a and a-b-c-g-a are also form two loops. Mesh- a mesh is a special case of loop that does not have any other loops within it or in its interior. Fig. 3.4 indicates that the first three loops (a-b-g-a; b-c-g-b; and a-c-b-a) just identified are also ‘meshes’ but other two loops (a-c-g-a and a-b-c-g- a) are not. With the introduction of the Kirchhoff’s laws, a various types of electric circuits can be analyzed. Kirchhoff’s Voltage Law or Kirchhoff’s loop rule Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 17 Basic Electrical Engineering Class notes Fig 3.2: Circuit diagram for the KVL This law is also called Kirchhoff’s loop rule. It is a consequence of the principle of conservation of energy. It states that “The algebraic sum of the potential differences around a circuit must be zero". Considering that electric potential is defined as line integral of electric field, Kirchhoff's voltage law can be expressed equivalently with equation Kirchhoff’s Current Law (KCL) Fig 3.3: Circuit diagram for the KCL This law is also called Kirchhoff's junction rule or Kirchhoff's point rule. This law states that “At any point in an electrical circuit where charge density is not changing in time, the sum of currents flowing towards that point is equal to the sum of currents flowing away from that point. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 18 Basic Electrical Engineering Class notes A charge density changing in time would mean the accumulation of a net positive or negative charge, which typically cannot happen to any significant degree because of the strength of electrostatic forces: the charge buildup would cause repulsive forces to disperse the charges. KCL states that at any node (junction) in a circuit the algebraic sum of currents entering and leaving a node at any instant of time must be equal to zero. Here currents entering (+ve sign) and currents leaving (-ve sign) the node must be assigned opposite algebraic signs. Fig 3.2: Illustration of Kirchhoff’s laws Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 19 Basic Electrical Engineering Class notes This law is also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule. The principle of conservation of energy implies that The directed sum of the electrical potential differences (voltage) around any closed network is zero, or: More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop, or: The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop. Similarly to KCL, it can be stated as: Here, n is the total number of voltages measured. The voltages may also be complex: This law is based on the conservation of energy whereby voltage is defined as the energy per unit charge. The total amount of energy gained per unit charge must be equal to the amount of energy lost per unit charge, as energy and charge are both conserved. Example: Assume an electric network consisting of two voltage sources and three resistors. According to the first law we have The second law applied to the closed circuit s1 gives The second law applied to the closed circuit s2 gives Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 20 Basic Electrical Engineering Class notes Thus we get a linear system of equations in : Assuming the solution is has a negative sign, which means that the direction of direction defined in the picture). is opposite to the assumed direction (the 1. Solve the circuit by nodal analysis and find Va . Solution a) Choose a reference node, label node voltages: Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 21 Basic Electrical Engineering Class notes b) Apply KCL to each node: Node 1: −Is2+ (V1−V2 ) /R3+(V1−V3) /R1 = 0 → 6V1−V2−5V =5 Node 2: Is3+V2−V1R3+V2R2=0→−4V1+9V2=−40 (1) (2) Node 3: −Is1−Is3+(V3−V1 ) /R1=0→V3−V1=4 (1), (2) and (3) imply that V1=37V,V2=12V and V3=41V . (3) c) Find the required quantities: If we apply KVL in the loop shown above: −V1−Va+V2=0→Va=−25V. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 22 Basic Electrical Engineering Class notes Practical Voltage and Current Sources Ideal and Practical Voltage Sources An ideal voltage source, which is represented by a model, is a device that produces a constant voltage across its terminals (V=E) no matter what current is drawn from it (terminal voltage is independent of load (resistance) connected across the terminals). Fig 5.1: Ideal DC voltage source For the circuit shown in fig, the upper terminal of load is marked plus (+) and its lower terminal is marked minus (-). This indicates that electrical potential of upper terminal is volts higher than that of lower terminal. The current flowing through the load RL is given by the expression VS =VL =ILRL and we can represent the terminal V−I characteristic of an ideal dc voltage as a straight line parallel to the x-axis. This means that the terminal voltage VL remains constant and equal to the source voltage VS irrespective of load current is small or large. The V−I characteristic of ideal voltage source is presented in Figure. However, real or practical dc voltage sources do not exhibit such characteristics in practice. We observed that as the load resistance RL connected across the source is decreased, the corresponding load current IL increases while the terminal voltage across the source decreases (see eq.3.1). We can realize such voltage drop across the terminals with increase in load current provided a resistance element (Rs) present inside the voltage source. Figshows the model of practical or real voltage source of value Vs. Fig 5.2: V-I characteristics of an Ideal voltage source The terminal V− I characteristics of the practical voltage source can be described by an equation VL = Vs - ILRs and this equation is represented graphically as shown in fig.3.16. In practice, when a load resistance RL more than 100 times larger than the source resistance Rs, the source can be considered approximately Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 23 Basic Electrical Engineering Class notes ideal voltage source. In other words, the internal resistance of the source can be omitted. This statement can be verified using the relation RL = 100 Rs in equation. The practical voltage source is characterized by two parameters namely known as (i) Open circuit voltage (Vs) (ii) Internal resistance in the source’s circuit model. In many practical situations, it is quite important to determine the source parameters experimentally. We shall discuss briefly a method in order to obtain source parameters. Fig 5.3: Practical dc voltage source Method-: Connect a variable load resistance across the source terminals (see fig.). A voltmeter is connected across the load and an ammeter is connected in series with the load resistance. Voltmeter and Ammeter readings for several choices of load resistances are presented on the graph paper (see fig. 5.4). The slope of the line is −Rs, while the curve intercepts with voltage axis (at IL = 0) is the value of Vs. Fig 5.4: V−I characteristic Practical dc voltage source The V−I characteristic of the source is also called the source’s “regulation curve” or “load line”. The open-circuit voltage is also called the “no-load” voltage, Voc. The maximum allowable load current (rated current) is known as full-load current IFl and the corresponding source or load terminal voltage is known as “full-load” voltage. We know that the source terminal voltage varies as the load is varied and this is due to internal voltage drop inside the source. The percentage change in source terminal voltage from no-load to full-load current is termed the “voltage regulation” of the source. It is defined as Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 24 Basic Electrical Engineering Class notes For ideal voltage source, there should be no change in terminal voltage from no-load to full-load and this corresponds to “zero voltage regulation”. For best possible performance, the voltage source should have the lowest possible regulation and this indicates a smallest possible internal voltage drop and the smallest possible internal resistance. Example: - A practical voltage source whose short-circuit current is 1.0A and open-circuit voltage is 24 Volts. What is the voltage across, and the value of power dissipated in the load resistance when this source is delivering current 0.25A? Solution: From fig. (Short-circuit test) (Open-circuit test) Therefore, the value of internal source resistance is obtained as Let us assume that the source is delivering current IL = 0.25A when the load resistance RL is connected across the source terminals. Mathematically, we can write the following expression to obtain the load resistance RL. . Now, the voltage across the load RL = 0.25 × 72 =18 volts. And the power consumed by the load is given by PL = IL× RL =0.625 × 72 = 4.5 watts Example- A certain voltage source has a terminal voltage of 50 V when I= 400 mA; when I rises to its full-load current value 800 mA the output voltage is recorded as 40 V. Calculate (i) Internal resistance of the voltage source (Rs). (ii) No-load voltage (open circuit voltage Vs). (iii) The voltage Regulation. Solution- From equation (VL = Vs - ILRs) one can write the following expressions under different loading conditions. 50 = Vs − 0.4Rs & 40 = Vs − 0.8Rs → solving these equations we get Vs = 60V & Rs =25 Ω = 60 40 100 33.33% 60 Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 25 Basic Electrical Engineering Class notes Ideal and Practical Current Sources There are several voltage sources as well as current sources encountered in our daily life. Batteries , DC generator or alternator all are very common examples of voltage source. There are also some current sources encountered in our everyday life, such as photo electric cells, metadynegenerator etc. The sources can be categorized into two different types – independent source and dependent source. • Another two-terminal element of common use in circuit modeling is current source` as depicted in fig.3.17. An ideal current source, which is represented by a model in fig., is a device that delivers a constant current to any load resistance connected across it, no matter what the terminal voltage is developed across the load (i.e., independent of the voltage across its terminals across the terminals). Fig 5.5: Ideal current source with variable load It can be noted from model of the current source that the current flowing from the source to the load is always constant for any load resistance (see fig.) i.e. whether R L is small (VL is small) or RL is large (VL is large). The vertical dashed line in fig. 5.6 represents the V-I characteristic of ideal current source. Fig 5.6: V-I characteristic of ideal current source Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 26 Basic Electrical Engineering Class notes Fig 5.7: Practical current source with variable load In practice when a load is connected across a practical current source, one can observe that the current flowing in load resistance is reduced as the voltage across the current source’s terminal is increased, by increasing the load resistance RL. Since the distribution of source current in two parallel paths entirely depends on the value of external resistance that connected across the source (current source) terminals. This fact can be realized by introducing a parallel resistance R s in parallel with the practical current source Is, as shown in fig. 5.7. The dark lines in fig.5.6 show the V-I characteristic (load-line) of practical current source. The slope of the curve represents the internal resistance of the source. One can apply KCL at the top terminal of the current source in fig. 5.7 to obtain the following expression. The open circuit voltage and the short-circuit current of the practical current source are given by VOC = ISRS and Ishort = Is respectively. It can be noted from the fig.5.6 that source 1 has a larger internal resistance than source 2 and the slope the curve indicates the internal resistance R s of the current source. Thus, source 1 is closer to the ideal source. More specifically, if the source internal resistance Rs ≥100 RL then source acts nearly as an ideal current source. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 27 Basic Electrical Engineering Class notes Independent Voltage Source Output of an independent source does not depend upon the voltage or current of any other part of the network. When terminal voltage of a voltage source is not affected by the current or voltage of any other part of the network, then the source is said to be an independent voltage source . This type of sources may be referred as constant source or time variant source. When terminal voltage of an independent source remains constant throughout its operation, it is referred as time–invariant or constant independent voltage source . Again independent voltage source can be time–variant type, where the output terminal voltage of the source changes with time. Here, the terminal voltage of the source does not vary with change of voltage or current of any other part of the network but it varies with time. Independent Current Source Similarly, output current of independent current source does not depend upon the voltage or current of any other part of the network. It is also categorized as independent timeinvariant and time-variant current source . Symbolic representations of independent time-invariant and time-variant voltage and current source s are shown below. Fig 6.1: Different types of independent sources Now we will discuss about dependent voltage or current source. Dependent voltage source is one that’s output voltage is the function of voltage or current of any other part of the circuit. Similarly, dependent current source is one that’s output current is the function of current or voltage of any other Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 28 Basic Electrical Engineering Class notes parts of the circuit. The amplifier is an ideal example of dependent source where the output signal depends upon the signal given to the input circuit of the amplifier. Dependent Voltage Source & Dependent Current Source There are four possible dependent sources as are represented below, 1. Voltage dependent voltage source . 2. Current dependent voltage source . 3. Voltage dependent current source . 4. Current dependent current source . Dependent voltage source s and dependent current sources can also be time variant or time invariant. That means, when the output voltage or current of a dependent source is varied with time, referred as time invariant dependent current or voltage source and if not varied with time, it is referred as time variant. a. VCVS b. ICVS c. VCIS d. ICIS Fig 6.2: different types of ideal dependent sources Ideal Voltage Source In every practical voltage source , there is some electrical resistance inside it. This resistance is called internal resistance of the source. When the terminal of the source is open circuited, there is no current flowing through it; hence there is no voltage drop inside the source but when load is connected with the source, current starts flowing through the load as well as the source itself. Due to the resistance inside the voltage source , there will be some voltage drop across the source. Now if any Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 29 Basic Electrical Engineering Class notes one measures the terminal voltage of the source, he or she will get the voltage between its terminals which is reduced by the amount of internal voltage drop of the source. So there will be always a difference between no-load (when source terminals are open) and load voltages of a practical voltage source . But in ideal voltage source this difference is considered as zero that means there would not be any voltage drop in it when current flows through it and this implies that the internal resistance of an ideal source must be zero. This can be concluded that, voltage across the source remains constant for all values of load current. The V-I characteristics of an ideal voltage source is shown below. Fig 6.3: VI Characteristics of Ideal Voltage Source There is no as such example of ideal voltage source but a lead acid battery or a dry cell can be considered an example when the current drawn is below a certain limit. Ideal Current Source Ideal current sources are those sources that supply constant current to the load irrespective of their impedance. That means, whatever may be the load impedance; ideal current source always gives same current through it. Even if the load has infinite impedance or load, is open circuited to the ideal current source that gives the same current through it. So naturally from definition, it is clear that this type of current source is not practically possible. Current Source to Voltage Source Conversion All sources of electrical energy give both current as well as voltage. This is not practically possible to distinguish between voltage source and current source . Any electrical source can be represented as voltage source as well as current source . It merely depends upon the operating condition. If the load impedance is much higher than internal impedance of the source, then it is Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 30 Basic Electrical Engineering Class notes preferable to consider the source as a voltage source on the other hand if the load impedance is much lower than internal impedance of the source; it is preferable to consider the source as a current source . Current source to voltage source conversion or voltage source to current source conversion is always possible. Now we will discuss how to convert a current source into voltage source and vice-versa. Let us consider a voltage source which has no load terminal voltage or source voltage V and internal resistance r. Now we have to convert this to an equivalent current source . For that, first we have to calculate the current which might be flowing through the source if the terminal A and B of the voltage source were short circuited. That would be nothing but I = V / r. This current will be supplied by the equivalent current source and that source will have the same resistance connected across it. Fig 6.4: Voltage to Current source conversion Similarly a current source of output current I in parallel with resistance r can be converted into an equivalent voltage source of voltage V = Ir and resistance r connected in series with it. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 31 Basic Electrical Engineering Class notes Nodal Voltage Analysis In this method, we set up and solve a system of equations in which the unknowns are the voltages at the principal nodes of the circuit. From these nodal voltages the currents in the various branches of the circuit are easily determined. Steps in the nodal analysis method: 1. Count the number of principal nodes or junctions in the circuit. Call this number n. (A principal node or junction is a point where 3 or more branches join. We will indicate them in a circuit diagram with a red dot. Note that if a branch contains no voltage sources or loads then that entire branch can be considered to be one node.) 2. Number the nodes N1, N2, . . . , Nn and draw them on the circuit diagram. Call the voltages at these nodes V1, V2, . . . , Vn, respectively. 3. Choose one of the nodes to be the reference node or ground and assign it a voltage of zero. 4. For each node except the reference node write down Kirchhoff’s Current Law in the form "the algebraic sum of the currents flowing out of a node equals zero". (By algebraic sum we mean that a current flowing into a node is to be considered a negative current flowing out of the node.) Express the current in each branch in terms of the nodal voltages at each end of the branch using Ohm's Law (I = V / R). Here are some examples: Fig 7.1: Nodal voltage analysis for given branches Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 32 Basic Electrical Engineering Class notes The result, after simplification, is a system of m linear equations in the m unknown nodal voltages (where m is one less than the number of nodes; m = n - 1). The equations are of this form: where G11, G12, . . . , Gmm and I1, I2, . . . , Im are constants. Example : Fig 7.2: Circuit diagram for analysis of the theorem in the example Number of nodes is 4. We will number the nodes as shown above. We will choose node 2 as the reference node and assign it a voltage of zero. Write down Kirchhoff’s Current Law for each node. Call V1 the voltage at node 1, V3 the voltage at node 3, V4 the voltage at node 4, and remember that V2 = 0. The result is the following system of equations: Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 33 Basic Electrical Engineering Class notes The first equation results from KCL applied at node 1, the second equation results from KCL applied at node 3 and the third equation results from KCL applied at node 4. Solving the above system of equations using Gaussian elimination or some other method gives the following voltages: V1 = -35.88 volts, V3 = 63.74 volts and V4 = 0.19 volts Mesh or Loop Analysis for Electric Circuits This is also called Loop Analysis. This method uses simultaneous equations, Kirchhoff's Voltage Law, and Ohm's Law to determine unknown currents in a network. Mesh Analysis only works for planar circuits: circuits that can be drawn on a plane (like on a paper) without any elements or wires crossing each other. In some cases a circuit that looks non-planar can be made in to a planar circuit by moving some of the connecting wires. The first step in the mesh Current method is to identify "loops" within the circuit encompassing all components. Represent all the loops with different loop currents in one direction. The choice of each loop current's direction is entirely arbitrary. The next step is to label all voltage drop polarities across resistors according to the assumed directions of the mesh currents. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 34 Basic Electrical Engineering Class notes Next write the KVL equations for each mesh and solve all the equations for mesh (loop) currents. Example: Fig 7.3: Circuit diagram for analysis of the theorem in the example KVL equation for Loop1: -28 + 2(I1+I2) + 4×I1 =0. KVL equation for Loop2: -2(I1+I2) + 7 - 1I2=0. Solving these 2 equations we get I1=5A I2=-1A. The solution of -1 amp for I2 means that our initially assumed direction of current was incorrect. In actuality, I2 is flowing in a counter-clockwise direction at a value of (positive) 1 amp. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 35 Basic Electrical Engineering Class notes Superposition Theorem The superposition theorem for electric circuits states that the total current in any branch of a bilateral linear circuit equals the algebraic sum of the currents produced by each source acting separately throughout the circuit. To ascertain the contribution of each individual source, all of the other sources first must be "killed" (set to zero) by: replacing all other voltage sources with a short circuit (thereby eliminating difference of potential. i.e. V=0) replacing all other current sources with an open circuit (thereby eliminating flow of current. i.e. I=0) This procedure is followed for each source in turn, and then the resultant currents are added to determine the true operation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources. Example: Fig 8.1: Circuit Diagram for analysis of the superposition theorem Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 36 Basic Electrical Engineering Class notes Fig 8.2: Circuit Diagram for application of the superposition theorem Fig 8.3: Circuit Diagram for analyzing of the superposition theorem for only 7V Fig 8.4: Circuit Diagram for analyzing of the superposition theorem for only 28V When superimposing these values of voltage and current, we have to be very careful to consider polarity (voltage drop) and direction (electron flow), as the values have to be added algebraically. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 37 Basic Electrical Engineering Class notes Fig 8.5: Circuit after superimposing voltage Fig 8.6: Circuit after superimposing voltage Thus the total voltage or current can be obtained in the circuit by just recombining the circuits with different sources. Another point that should be considered is that superposition only works for voltage and current but not power. In other words the sum of the powers of each source with the other sources turned off is not the real consumed power. To calculate power we should first use superposition to find both current and voltage of each linear element and then calculate the sum of the multiplied voltages and currents. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 38 Basic Electrical Engineering Class notes Thevenin's Theorem: This theorem was first discovered by German scientist Hermann von Helmholtz in 1853, but was then rediscovered in 1883 by French telegraph engineer Léon Charles Thévenin (1857-1926). Fig 9.1: Circuit diagram for conversion of any given circuit to a Thevenin’s Equivalent circuit Any black box containing only voltage source, current source or resisters can be converted to Thevenin’s Equivalent circuit. This theorem states that any linear bilateral circuit with combination of voltage sources , current sources and resistors with two terminals is electrically equivalent to a single voltage source V Th (Thevenin’s Voltage) and a single series resistor RTh (Thevenin’s Resistance).This equivalent is called Thevenin’s Equivalent. Calculating the Thevenin’s equivalent: To calculate the equivalent circuit, one needs a resistance and a voltage - two unknowns. And so, one needs two equations. These two equations are usually obtained by using the following steps, but any conditions one places on the terminals of the circuit should also work: 1. Calculate the output voltage, VAB, when in open circuit condition (no load resistor - meaning infinite resistance). This is VTh. 2. Calculate the output current, IAB, when those leads are short circuited (load resistance is 0) RTh equals VTh divided by this IAB. Case 2 could also be thought of like this: 2a. Now replace voltage sources with short circuits and current sources with open circuits. 2b. Replace the load circuit with an imaginary ohm meter and measure the total resistance, R, "looking back" into the circuit. This is RTh. The Thevenin’s-equivalent voltage is the voltage at the output terminals of the original circuit. When calculating a Thevenin’s-equivalent voltage, the voltage divider principle is often useful, by declaring one terminal to be Vout and the other terminal to be at the ground point. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 39 Basic Electrical Engineering Class notes The Thevenin’s-equivalent resistance is the resistance measured across points A and B "looking back" into the circuit. It is important to first replace all voltage- and current-sources with their internal resistances. For an ideal voltage source, this means replace the voltage source with a short circuit. For an ideal current source, this means replace the current source with an open circuit. Resistance can then be calculated across the terminals using the formulae for series and parallel circuits. A Norton equivalent circuit is related to the Thevenin’s equivalent by the following equations: RTh = RNo , VTh = INo RNo Example of a Thevenin’s equivalent circuit Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 40 Basic Electrical Engineering Class notes In the example, calculating equivalent voltage and resistance: Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 41 Basic Electrical Engineering Class notes Norton's Theorem : Norton's theorem is an extension of Thevenin's theorem and was introduced in 1926 separately by two people: Hause-Siemens researcher Hans Ferdinand Mayer (1895-1980) and Bell Labs engineer Edward Lawry Norton (1898-1983). Mayer was the only one of the two who actually published on this topic, but Norton made known his finding through an internal technical report at Bell Labs. Fig 10.1: Circuit diagram for conversion of any given circuit to a Norton's Equivalent circuit Any black box containing only voltage source, current source or resisters can be converted to Norton's Equivalent circuit. This theorem states that any linear bilateral circuit with combination of voltage sources, current sources and resistors with two terminals is electrically equivalent to an ideal current soure, INo, in parallel with a single resistor, RNo. This equivalent is called Norton Equivalent. Calculation of Norton Equivalent To calculate the equivalent circuit: 1. Calculate the output current, IAB, when a short circuit is the load (meaning 0 resistance between A and B). This is INo. 2. Calculate the output voltage, VAB, when in open circuit condition (no load resistor - meaning infinite resistance). RNo equals this VAB divided by INo. The equivalent circuit is a current source with current INo, in parallel with a resistance RNo. Case 2 can also be thought of like this: 2a. Now replace independent voltage sources with short circuits and independent current sources with open circuits. Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 42 Basic Electrical Engineering Class notes 2b. For circuits without dependent sources RNo is the total resistance with the independent sources removed. * Note: A more general method for determining the Norton Impedance is to connect a current source at the output terminals of the circuit with a value of 1 Ampere and calculate the voltage at its terminals; this voltage is equal to the impedance of the circuit. This method must be used if the circuit contains dependent sources. This method is not shown below in the diagrams. To convert to a Thevenin’s equivalent circuit, one can use the following equations: RTh = RNo VTh = INo RNo Example of Norton Equivalent circuit Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 43 Basic Electrical Engineering Class notes Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 44 Basic Electrical Engineering Class notes Maximum Power Transfer Theorem This theorem was first discovered by Moritz von Jacobi which is referred to as "Jacobi’s law". The theorem states that in a circuit maximum power is transferred from source to load when the resistance of the load is same as that of the source. The theorem applies only when the source resistance is fixed. If the source resistance were variable (but the load resistance fixed), maximum power would be transferred to the load simply by setting the source resistance to zero. Raising the source impedance to match the load would, in this case, reduce power transfer. This is the case when driving a load such as a loudspeaker with a modern amplifier. In this case, the load presented by the loudspeaker is fixed (typically, 8 ohms for home audio) and maximum power occurs with an impedance bridging connection. This type of connection also serves to maximize control of the speaker cone (due to high damping factor), which serves to lower distortion. It is important to note that maximum efficiency is not the same as maximum power transfer. To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be made close to zero. Fig 11.1: circuit for analysis of Maximum power Transfer The condition of maximum power transfer does not result in maximum efficiency. If we define the efficiency η as the ratio of power dissipated by the load to power developed by the source, then it is straightforward to calculate from the circuit diagram that Consider three particular cases : Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 45 Basic Electrical Engineering Class notes If RLoad = RSource then η = 0.5 If RLoad = infinity then η = 1 If RLoad = 0 then η = 0 The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the load resistance approaches infinity (though the total power level tends towards zero). When the load resistance is zero, all the power is consumed inside the source (the power dissipated in a short circuit is zero) so the efficiency is zero. Voltage Source : When a load resistance RL is connected to a voltage source VS with series resistance RS, maximum power transfer to the load occurs when RL is equal to RS. Under maximum power transfer conditions, the load resistance R L, load voltage VL, load current IL and load power PL are: RL = RS VL = IL = VS 2 VS VL = 2 RS RL PL = V L × VS VL = VS × 4 RS RL Current Source: Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 46 Basic Electrical Engineering Class notes When a load conductance GL is connected to a current source IS with shunt conductance GS, maximum power transfer to the load occurs when GL is equal to GS. Under maximum power transfer conditions, the load conductance GL, load current IL, load voltage V L and load power PL are: GL = GS IL = VL = IS 2 IS IL = 2 GS GL PL = IL× IS IL = IS × 4 GS GL Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering 47 Basic Electrical Engineering Class notes Y-Δ transformation & Δ-Y transformation Delta-Star Transformation The Y-Δ transform, also written Y-delta, Wye-delta, Kennelly’s delta-star transformation, starmesh transformation, T-Π or T-pi transform, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. In the United Kingdom, the wye diagram is known as a star. The transformation is used to establish equivalence for networks with 3 terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for real as well as complex impedances. Equations for the transformation from Δ-load to Y-load 3-phase circuit the general idea is to compute the impedance Ry at a terminal node of the Y circuit with impedances where RΔ R', are all R'' to impedances adjacent in the Δ nodes in the circuit. This yields Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering Δ the circuit specific by formulae 48 Basic Electrical Engineering Class notes Equations for the transformation from Y-load to Δ-load 3-phase circuit The general idea is to compute an impedance RΔ in the Δ circuit by Where Rp = R1R2 + R2R3 + R3R1 is the sum of the products of all pairs of impedances in the Y circuit and Ropposite is the impedance of the node in the Y circuit which is opposite the edge with RΔ. The formulae for the individual Mr.Manmohan Panda (Lecturer) Dept of Electrical Engineering edges are thus 49