2.1 Resistors. Ohm’s LawFigure 2-1 Circuit symbol of a resistorOhm’s law can also be written as2.3 Open and Short CircuitFigure 2-3 a) Short-circuit symbol, b) open circuit symbolFigure 2-4 Ideal switch symbolFigure 2-7 i-v characteristic of an ideal voltage sourceFigure 2-9 i-v characteristic of an ideal current source2.7 Dependent SourcesSymbols for dependent sources are shown in Figure 2-10.2.8 Kirchhoff’s Current LawExample 2.1Equation (2.14) may be rearranged in the formFigure 2-12 A single-loop circuit illustrating KVLFigure 2-13 Voltage sources in series: (a) original circuit,Figure 2-14 Circuit for Example 2.2Figure 2-15 Circuit with assigned reference directions2.11 Voltage Divider2.12 Series Equivalent Resistance2.13 Current DividerConsider the circuit in Fig. 4-3, where two resistors are coFigure 4-3 Two resistors in parallelFigure 4-5 Resistor parallel with a short circuitFigure 4-6 Resistor parallel with an open circuitExample 4-1Figure 4-7 Circuit for Example 4-12.15 Ladder CircuitsCircuit Analysis I II-1 Basic Laws Prof. Bogdan Adamczyk Grand Valley State University Winter 2008 CHAPTER 2 FUNDAMENTAL CIRCUIT ANALYSIS METHODS We have, thus far, introduced basic concepts such as current, voltage, power and energy in an electric circuit. To actually determine the values of these variables in a given circuit requires that we understand some fundamental laws that govern electric circuits. These laws, known as Ohm’s law and Kirchhoff’s laws, form the foundation upon which electric circuit analysis is built. Any electrical circuit consists of basic building blocks called circuit elements. Any two-terminal element is described by the relationship between the current flowing through it and the voltage across it, the so-called i-v characteristics. This i-v characteristics for any circuit element is also called an element constraint, since it constraints the current and voltage to satisfy a given relation and not to have arbitrary values. In most cases this relationship is complicated and nonlinear, so we use a simplified linear model. 2.1 Resistors. Ohm’s Law Materials in general have the ability to resist the flow of electric charge. This physical property of a material, the ability to resist current, is known as resistance, and is represented by the symbol R. Resistance is the capacity of materials to impede the flow of current or, more specifically, the flow of electric charge. The circuit element used to model this behavior is the resistor. The circuit symbol of a resistor is shown in Fig. 2-1. The resistor R, is connected between two points (nodes) a nd b, has a current i flowing through it, and a voltage v across it. i + a R v − Figure 2-1 Circuit symbol of a resistor bCircuit Analysis I II-2 Basic Laws Prof. Bogdan Adamczyk Grand Valley State University Winter 2008 German physicist Georg Simon Ohm (1787-1854) is credited with finding the relationship between current and voltage for a resistor. This relationship is known as Ohm’s law. Ohm’s law states that the voltage across a resistor is directly proportional to the current flowing through the resistor. The constant of proportionality is the resistance value of the resistor. Mathematical form of the Ohm’s law is iRv = (2.1) R is measures in units called ohms, that can be obtained from Eq. (2.2) ⎥⎦⎤⎢⎣⎡==ΩAVivR (2.2) The equation (2.2) is an i-v characteristics for a linear resistor. It is also an element constraint. Fig. 2-2 shows the graphical representation of this i-v characteristics. v α i Rivtanslope ===α Figure 2-2 i-v characteristics of a linear resistor To apply Ohm’s law, as stated in Eq. (2.1), we must pay close attention to the voltage and current reference directions. When using Eq. (2.1), the voltage and current must conform to the passive sign convention. If they do not, the Ohm’s law assumes the form iRv −= (2.3)Circuit Analysis I II-3 Basic Laws Prof. Bogdan Adamczyk Grand Valley State University Winter 2008 2.2 Conductance Ohm’s law can also be written as vGi = (2.4) where G denotes the conductance in siemens (S) and is the reciprocal of R. ⎥⎦⎤⎢⎣⎡===VA1SR1GΩ (2.5) Conductance is a measure of how well an element will conduct electric current. ♦ Using Ohm’s law, the power delivered to the resistor (or dissipated by the resistor) can be expressed as ⎥⎦⎤⎢⎣⎡======2222VAVAWRviRivpΩΩ (2.6) where v and i have been assumed to satisfy the passive sign convention. Using the definition of conductance, the power dissipated by the resistor can alternatively be expressed as 22GvGivip === (2.7)Circuit Analysis I II-4 Basic Laws Prof. Bogdan Adamczyk Grand Valley State University Winter 2008 2.3 Open and Short Circuit The value of a resistance R can range from zero to infinity. We will now consider these two important extreme cases. An element with is called a short circuit, and is shown in Fig. 2-3(a) 0R = i i + + v v R = ∞ R = 0 − − a) b) Figure 2-3 a) Short-circuit symbol, b) open circuit symbol According to Ohm’s law, for a short circuit, 0iRv == (2.8) showing that a voltage across a short circuit is zero. In practice, a short circuit corresponds to a connecting wire, which has very low resistance. An element with ∞=R is called an open circuit, and is shown in Fig. 2-3 (b). According to Ohm’s law, for an open circuit 0Rvi == (2.9) Showing that a current through an open circuit is zero. In practice, an open circuit is equivalent to a break in the wire.Circuit Analysis I II-5 Basic Laws Prof. Bogdan Adamczyk Grand Valley State University Winter 2008 2.4 Ideal Switch The ideal switch can be modeled as a combination of an open- and short-circuit elements. Fig. 2-4 shows the circuit symbol of an ideal switch. i i + v OFF (open) ON (closed) + v − − Figure 2-4 Ideal switch symbol When the switch is open , and when it is closed 0i = 0v=. 2.5 Ideal Independent Voltage Source The power sources required to operate electric circuits are modeled using two elements: voltage sources and current sources. Before discussing ideal voltage and current sources, we need to consider the general nature of electrical sources. An electrical source is a device that is capable of converting nonelectric energy to electric energy and vice