CHAPTER 2 Basic Laws Here we explore two fundamental laws that govern electric circuits (Ohm's law and Kirchhoff's laws) and discuss some techniques commonly applied in circuit design and analysis. 2.1. Ohm's Law Ohm's law shows a relationship between voltage and current of a resis- tive element such as conducting wire or light bulb. 2.1.1. Ohm's Law: The voltage v across a resistor is directly propor- tional to the current i flowing through the resistor. v = iR; where R = resistance of the resistor, denoting its ability to resist the flow of electric current. The resistance is measured in ohms (Ω). • To apply Ohm's law, the direction of current i and the polarity of voltage v must conform with the passive sign convention. This im- plies that current flows from a higher potential to a lower potential 15 16 2. BASIC LAWS in order for v = iR. If current flows from a lower potential to a higher potential, v = −iR. l i + v R – Material with Cross-sectional resistivity r area A 2.1.2. The resistance R of a cylindrical conductor of cross-sectional area A, length L, and conductivity σ is given by L R = : σA Alternatively, L R = ρ A where ρ is known as the resistivity of the material in ohm-meters. Good conductors, such as copper and aluminum, have low resistivities, while insulators, such as mica and paper, have high resistivities. 2.1.3. Remarks: (a) R = v=i (b) Conductance : 1 i G = = R v 1 2 The unit of G is the mho (f) or siemens (S) 1Yes, this is NOT a typo! It was derived from spelling ohm backwards. 2In English, the term siemens is used both for the singular and plural. 2.1. OHM'S LAW 17 (c) The two extreme possible values of R. (i) When R = 0, we have a short circuit and v = iR = 0 showing that v = 0 for any i. + i v = 0 R = 0 – (ii) When R = 1, we have an open circuit and v i = lim = 0 R!1 R indicating that i = 0 for any v. + i = 0 v R = ∞ – 2.1.4. A resistor is either fixed or variable. Most resistors are of the fixed type, meaning their resistance remains constant. 18 2. BASIC LAWS A common variable resistor is known as a potentiometer or pot for short 2.1.5. Not all resistors obey Ohms law. A resistor that obeys Ohms law is known as a linear resistor. • A nonlinear resistor does not obey Ohms law. • Examples of devices with nonlinear resistance are the lightbulb and the diode. • Although all practical resistors may exhibit nonlinear behavior un- der certain conditions, we will assume in this class that all elements actually designated as resistors are linear. 2.1. OHM'S LAW 19 2.1.6. Using Ohm's law, the power p dissipated by a resistor R is v2 p = vi = i2R = : R Example 2.1.7. In the circuit below, calculate the current i, and the power p. i + 30 V DC 5 kΩ v – Definition 2.1.8. The power rating is the maximum allowable power dissipation in the resistor. Exceeding this power rating leads to overheating and can cause the resistor to burn up. Example 2.1.9. Determine the minimum resistor size that can be con- 1 nected to a 1.5V battery without exceeding the resistor's 4-W power rating. 54 CHAPTER TWO resistive networks illustrative circuit. We will then apply the same systematic method to solve more complicated examples, including the one shown in Figure 2.1. 202.1 TERMINOLOGY 2. BASIC LAWS Lumped circuit elements are the fundamental building blocks of electronic cir- cuits. Virtually2.2. all of Node, our analyses Branches will be conducted and Loops on circuits containing two-terminal elements; multi-terminal elements will be modeled using combi- Definitionnations of two-terminal2.2.1. Since elements. the elements We have already of an seen electric several two-terminalcircuit can be in- terconnectedelements in such several as resistors, ways, voltage we need sources, to and understand current sources. some Electronic basic concept access to an element is made through its terminals. of networkAn topology. electronic circuit is constructed by connecting together a collection of • Networkseparate elements = interconnection at their terminals, as of shown elements in Figure or 2.2. devices The junction points • Circuitat which the=a terminals network of two with or more closed elements paths are connected are referred to as the nodes of a circuit. Similarly, the connections between the nodes are referred Definitionto as the edges2.2.2 or. branchesBranchof: a A circuit. branch Note represents that each element a single in Figure element 2.2 such forms a single branch. Thus an element and a branch are the same for circuits as a voltagecomprising source only two-terminalor a resistor. elements. A branch Finally, circuit representsloops are defined any two-terminal to be element.closed paths through a circuit along its branches. Several nodes, branches, and loops are identified in Figure 2.2. In the circuit Definitionin Figure 2.2,2.2.3 there. Node are 10 branches: A node (and is thus, the 10 \point" elements) of and connection 6 nodes. between two or moreAs branches. another example, a is a node in the circuit depicted in Figure 2.1 at • Itwhich is usually three branches indicated meet. Similarly, by a dotb is in a node a circuit. at which two branches meet. ab and bc are examples of branches in the circuit. The circuit has five branches • Ifand a four short nodes. circuit (a connecting wire) connects two nodes, the two nodesSince constitute we assume that a single the interconnections node. between the elements in a circuit are perfect (i.e., the wires are ideal), then it is not necessary for a set of elements Definitionto be joined2.2.4 together. Loop at a single: A point loop in isspace any for closed their interconnection path in a to circuit. be A closed pathconsidered is formed a single by node. starting An example at a of node, this is passing shown in through Figure 2.3. a While set of nodes and returningthe four to elements the starting in the figure node are connected without together, passing their through connection any does node more not occur at a single point in space. Rather, it is a distributed connection. than once. Nodes Loop Elements FIGURE 2.2 An arbitrary circuit. Branch Definition 2.2.5. Series: Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current. Definition 2.2.6. Parallel: Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them. 2.2. NODE, BRANCHES AND LOOPS 21 2.2.7. Elements may be connected in a way that they are neither in series nor in parallel. Example 2.2.8. How many branches and nodes does the circuit in the following figure have? Identify the elements that are in series and in par- allel. 5 Ω 1 Ω 2 Ω DC 10 V 4 Ω 2.2.9. A loop is said to be independent if it contains a branch which 2.2 Kirchhoff’s Laws CHAPTER TWO 55 is not in any other loop. Independent loops or paths result in independent sets of equations. A network with b branches, n nodes, and ` independent loops will satisfy the fundamental theorem of network topology: Elements B b = ` + n − 1: B FIGURE 2.3 Distributed C Definition 2.2.10. The primary signals withinA a circuitC are its currents interconnections of four circuit and voltages,A which we denote by the symbols i andD v, respectively. We elements that nonetheless occur define a branch currentD asDistributed the current node along a branch of the circuit, and at a single node. Ideal wires a branch voltage as the potential difference measured across a branch. - FIGURE 2.4 Voltage and current i v definitions illustrated on a branch in Branch a circuit. current Branch + voltage Nonetheless, because the interconnections are perfect, the connection can be considered to be a single node, as indicated in the figure. The primary signals within a circuit are its currents and voltages, which we denote by the symbols i and v, respectively. We define a branch current as the current along a branch of the circuit (see Figure 2.4), and a branch voltage as the potential difference measured across a branch. Since elements and branches are the same for circuits formed of two-terminal elements, the branch voltages and currents are the same as the corresponding terminal variables for the elements forming the branches. Recall, as defined in Chapter 1, the terminal variables for an element are the voltage across and the current through the element. As an example, i4 is a branch current that flows through branch bc in the circuit in Figure 2.1. Similarly, v4 is the branch voltage for the branch bc. 2.2 KIRCHHOFF’S LAWS Kirchhoff’s current law and Kirchhoff’s voltage law describe how lumped- parameter circuit elements couple at their terminals when they are assembled into a circuit. KCL and KVL are themselves lumped-parameter simplifications of Maxwell’s Equations. This section defines KCL and KVL and justifies that they are reasonable using intuitive arguments.1 These laws are employed in circuit analysis throughout this book. 1. The interested reader can refer to Section A.2 in Appendix A for a derivation of Kirchhoff’s laws from Maxwell’s Equations under the lumped matter discipline.