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Introduction TO Elecrtical Engineering 2
Electrical engineering (EET301)
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FUNDAMENTALS OF ELECTRIC
CIRCUITS
Chapter 2 presents the fundamental laws that govern the behavior of electric circuits, and it serves as the foundation to the remainder of this book. The chapter beginswith a series of definitions to acquaint thereaderwith electric circuits; next, the two fundamental laws of circuit analysis are introduced: Kirchhoff’s current and voltage laws. With the aid of these tools, the concepts of electric power and the sign convention and methods for describing circuit elements—resistors in particular—are presented. Following these preliminary topics, the emphasis moves to basic analysis techniques—voltage and current dividers, and to some application examples related to the engineering use of these concepts. Examples include a description of strain gauges, circuits for the measurements of force and other related mechanical variables, and of the study of an automotive throttle position sensor. The chapter closes with a brief discussion of electricmeasuring instruments following box outlines the principal learning objectives of the chapter.
2 DEFINITIONS
In this section, we formally define some variables and concepts that are used in the remainder of the chapter. First, we define voltage and current sources; next, we define the concepts of branch, node, loop, and mesh, which form the basis of circuit analysis. Intuitively, an ideal source is a source that can provide an arbitrary amount of energy. Ideal sources are divided into two types: voltage sources and current sources. Of these, you are probably more familiar with the first, since dry-cell, alkaline, and lead-acid batteries are all voltage sources (they are not ideal, of course). You might have to think harder to come up with a physical example that approximates the behavior of an ideal current source; however, reasonably good approximations of ideal current sources also exist. For instance, a voltage source connected in series with a circuit element that has a large resistance to the flow of current from the source provides a nearly constant—though small—current and therefore acts very nearly as an ideal current source. A battery charger is another example of a device that can operate as a current source.
Ideal Voltage Sources An ideal voltage source is an electric device that generates a prescribed voltage at its terminals. The ability of an ideal voltage source to generate its output voltage is not affected by the current it must supply to the other circuit elements. Another way to phrase the same idea is as follows:
An ideal voltage source provides a prescribed voltage across its terminals irrespective of the current flowing through it. The amount of current supplied by the source is determined by the circuit connected to it.
Figure 2 depicts various symbols for voltage sources that are employed throughout this book. Note that the output voltage of an ideal source can be a function of time. In general, the following notation is employed in this book, unless otherwise noted. A generic voltage source is denoted by a lowercase v. If it is necessary to emphasize that the source produces a time-varying voltage, then the notation v(t) is employed. Finally, a constant, or direct current, or DC, voltage source is denoted by
the uppercase character V. Note that by convention the direction of positive current flow out of a voltage source is out of the positive terminal. The notion of an ideal voltage source is best appreciated within the context of the source-load representation of electric circuits. Figure 2 depicts the connection of an energy source with a passive circuit (i., a circuit that can absorb and dissipate energy). Three different representations are shown to illustrate the conceptual, symbolic, and physical significance of this source-load idea.
In the analysis of electric circuits, we choose to represent the physical reality of Figure 2(c) by means of the approximation provided by ideal circuit elements, as depicted in Figure 2(b). Ideal Current Sources An ideal current source is a device that can generate a prescribed current independent of the circuit to which it is connected. To do so, it must be able to generate an arbitrary voltage across its terminals. Figure 2 depicts the symbol used to represent ideal current sources. By analogy with the definition of the ideal voltage source just stated, we write that
An ideal current source provides a prescribed current to any circuit connected to it. to it. The voltage generated by the source is determined by the circuit connected.
rightmost circuit of Figure 2. Supernodes can be treated in exactly the same way as nodes.
Loop Ain Figure 2. loop is any closed connection of branches loop configurations are illustrated.
Mesh Acertain analysismethods. In Figure 2, the circuit with loops 1, 2, and 3 consists of two mesh is a loop that does not contain other loops. Meshes are an important aid to meshes: Loops 1 and 2 are meshes, but loop 3 is not a mesh, because it encircles both loops 1 and 2. The one-loop circuit of Figure 2 is also a one-mesh circuit. Figure 2. illustrates how meshes are simpler to visualize in complex networks than loops are.
Network Analysis The analysis of an electrical network consists of determining each of the unknown branch currents and node voltages. It is therefore important to define all the relevant variables as clearly as possible and in systematic fashion. Once the known and unknown variables have been identified, a set of equations relating these variables is constructed, and these are solved by means of suitable techniques. Before introducing methods for the analysis of electrical networks, we must formally present some important laws of circuit analysis.
2 CHARGE, CURRENT, AND KIRCHHOFF’S
CURRENT LAW
The earliest accounts of electricity date from about 2,500 years ago, when it was discovered that static charge on a piece of amber was capable of attracting very light objects, such as feathers. The word electricity originated about 600 B.; it comes from
The voltage, or potential difference, between two points in a circuit indicates the energy required to move charge from one point to the other. The role played by a voltage source in an electric circuit is equivalent to that played by the force of gravity. Raising a mass with respect to a reference surface increases its potential energy. This potential energy can be converted to kinetic energy when the object moves to a lower position relative to the reference surface. The voltage, or potential difference, across a voltage source plays an analogous role, raising the electrical potential of the circuit, so that charge can move in the circuit, converting the potential energy within the voltage source to electric power. As will be presently shown, the direction, or polarity, of the voltage is closely tied to whether energy is being dissipated or generated in the process. The seemingly abstract concept of work being done in moving charges can be directly applied to the analysis of electric circuits; consider again the simple circuit consisting of a battery and a lightbulb. The circuit is drawn again for convenience in Figure 2, with nodes defined by the letters a and b. Experimental observations led Kirchhoff to the formulation of the second of his laws, Kirchhoff’s voltage law, or KVL. The principle underlying KVL is that no energy is lost or created in an electric circuit; in circuit terms, the sum of all voltages associated with sources must equal the sum of the load voltages, so that the net voltage around a closed circuit is zero. If this were not the case, we would need to find a physical explanation for the excess (or missing) energy not accounted for in the voltages around a circuit. Kirchhoff’s voltage law may be stated in a form similar to that used for KCL.
where the vn are the individual voltages around the closed circuit. To understand this concept, we must introduce the concept of reference voltage.
In Figure 2, the voltage across the lightbulb is the difference between two node voltages, va and vb. In a circuit, any one node may be chosen as the reference node, such that all node voltages may be referenced to this reference voltage. In Figure 2, we could select the voltage at node b as the reference voltage and observe that the battery’s positive terminal is 1 V above the reference voltage. It is convenient to assign a value of zero to reference voltages, since this simplifies the voltage assignments around the circuit. With reference to Figure 2, and assuming Part I Circuits 21 that vb = 0, we can write v 1 = 1 V v 2 = vab = va - vb = va - 0 = va but va and v 1 are the same voltage, that is, the voltage at node a (referenced to node b). Thus v 1 = v 2 One may think of the work done in moving a charge from point a to point b and the work done moving it back from b to a as corresponding directly to the voltages across individual circuit elements. Let Q be the total charge that moves around the circuit per unit time, giving rise to current i. Then the work W done in moving Q from b to a (i., across the battery) is Wba = Q × 1 V Similarly, work is done in moving Q from a to b, that is, across the lightbulb. Note
that the word potential is quite appropriate as a synonym of voltage, in that voltage represents the potential energy between two points in a circuit: If we remove the lightbulb from its connections to the battery, there still exists a voltage across the (now disconnected) terminals b and a. This is illustrated in Figure 2.
A moment’s reflection upon the significance of voltage should suggest that it must be necessary to specify a sign for this quantity. Consider, again, the same drycell or alkaline battery where, by virtue of an electrochemically induced separation of charge, a 1-V potential difference is generated. The potential generated by the battery may be used to move charge in a circuit. The rate at which charge is moved once a closed circuit is established (i., the current drawn by the circuit connected to the battery) depends now on the circuit element we choose to connect to the battery. Thus, while the voltage across the battery represents the potential for providing energy to a circuit, the voltage across the lightbulb indicates the amount of work done in dissipating energy. In the first case, energy is generated; in the second, it is consumed (note that energy may also be stored, by suitable circuit elements yet to be introduced). This fundamental distinction requires attention in defining the sign (or polarity) of voltages. We shall, in general, refer to elements that provide energy as sources and to elements that dissipate energy as loads. Standard symbols for a generalized sourceand- load circuit are shown in Figure 2. Formal definitions are given later.
Ground The concept of reference voltage finds a practical use in the ground voltage of a circuit. Ground represents a specific reference voltage that is usually a clearly identified point in a circuit. For example, the ground reference voltage can be identified with the case or enclosure of an instrument, or with the earth itself. In residential electric circuits, the ground reference is a large conductor that is physically connected to the earth. It is convenient to assign a potential of 0 V to the ground voltage reference. The choice of the word ground is not arbitrary. This point can be illustrated by a simple analogy with the physics of fluid motion. Consider a tank of water, as shown in Figure 2, located at a certain height above the ground. The potential energy due to gravity will cause water to flow out of the pipe at a certain flow rate. The pressure that forces water out of the pipe is directly related to the head h 1 - h 2 in such a way that this pressure is zero when h 2 = h 1. Now the point h 3 , corresponding to the ground level, is defined as having zero potential energy. It should be apparent that the pressure acting on the fluid in the pipe is really caused by the difference in potential energy (h 1 - h 3 ) - (h 2 - h 3 ). It can be seen, then, that it is not necessary to assign a precise energy level to the height h 3 ; in fact, it would be extremely cumbersome to do so, since the equations describing the flow of water would then be different, say, in Denver, Colorado (h 3 = 1,600 m above sea level), from those
sign convention has been applied consistently, the answer will be correct regardless of the reference direction chosen. Examples 2 and 2 illustrate this point.
Introduction TO Elecrtical Engineering 2
Course: Electrical engineering (EET301)
University: Rashtrasant Tukadoji Maharaj Nagpur University
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