CHAPTER 2. Basic Laws

Transcription

1 CHAPTER 2 Basic Laws Here we explore two fundamental laws that goern electric circuits (Ohm s law and Kirchhoff s laws) and discuss some techniques commonly applied in circuit design and analysis Ohm s Law Ohm s law shows a relationship between oltage and current of a resistie element such as conducting wire or light bulb Ohm s Law: The oltage across a resistor is directly proportional to the current i flowing through the resistor. = 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 oltage must conform with the passie sign conention. This implies that current flows from a higher potential to a lower potential 15

2 16 2. BASIC LAWS in order for = ir. If current flows from a lower potential to a higher potential, = ir. l i Cross-sectional area A Material with resistiity r R The resistance R of a cylindrical conductor of cross-sectional area A, length L, and conductiity σ is gien by R = L σa. Alternatiely, R = ρ L A where ρ is known as the resistiity of the material in ohm-meters. Good conductors, such as copper and aluminum, hae low resistiities, while insulators, such as mica and paper, hae high resistiities Remarks: (a) R = /i (b) Conductance : G = 1 R = i The unit of G is the mho 1 ( ) or siemens 2 (S) 1 Yes, this is NOT a typo! It was deried from spelling ohm backwards. 2 In English, the term siemens is used both for the singular and plural.

3 2.1. OHM S LAW 17 (c) The two extreme possible alues of R. (i) When R = 0, we hae a short circuit and = ir = 0 showing that = 0 for any i. = 0 i R = 0 (ii) When R =, we hae an open circuit and i = lim R R = 0 indicating that i = 0 for any. i = 0 R = A resistor is either fixed or ariable. Most resistors are of the fixed type, meaning their resistance remains constant.

4 18 2. BASIC LAWS A common ariable resistor is known as a potentiometer or pot for short 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 deices with nonlinear resistance are the lightbulb and the diode. Although all practical resistors may exhibit nonlinear behaior under certain conditions, we will assume in this class that all elements actually designated as resistors are linear.

5 2.1. OHM S LAW Using Ohm s law, the power p dissipated by a resistor R is p = i = i 2 R = 2 R. Example In the circuit below, calculate the current i, and the power p. i 30 V DC 5 kω Definition The power rating is the maximum allowable power dissipation in the resistor. Exceeding this power rating leads to oerheating and can cause the resistor to burn up. Example Determine the minimum resistor size that can be connected to a 1.5V battery without exceeding the resistor s 1 4-W power rating.

6 illustratie circuit. We will then apply the same systematic method to sole more complicated examples, including the one shown in Figure TERMINOLOGY BASIC LAWS Lumped circuit elements are the fundamental building blocks of electronic circuits. Virtually2.2. ofnode, our analyses Branches will be conducted and Loops on circuits containing two-terminal elements; multi-terminal elements will be modeled using combi- Definition nations of two-terminal Since elements. the elements We hae already of anseen electric seeral two-terminal circuit can be interconnected in seeral ways, we need to understand some basic concept elements such as resistors, oltage sources, current sources. Electronic access to an element is made through its terminals. of network An topology. electronic circuit is constructed by connecting together a collection of Network separate elements = interconnection at their terminals, as ofshown elements in Figure or2.2. deices The junction points Circuit at which the = aterminals 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 to as the edges or branches of a circuit. Note that each element in Figure 2.2 Definition Branch: A branch represents a single element such forms a single branch. Thus an element and a branch are the same for circuits as a oltage comprising source only two-terminal or a resistor. elements. A branch Finally, circuit represents loops are defined any two-terminal to be element. closed paths through a circuit along its branches. Seeral nodes, branches, and loops are identified in Figure 2.2. In the circuit Definition Figure 2.2, therenode: are 10 branches A node (andis thus, the 10 point elements) of andconnection 6 nodes. between two or moreasbranches. another example, a is a node in the circuit depicted in Figure 2.1 at which three branches meet. Similarly, b is a node at which two branches meet. It is usually indicated by a dot in a circuit. ab and bc are examples of branches in the circuit. The circuit has fie branches Ifand a four short nodes. circuit (a connecting wire) connects two nodes, the two nodes Since 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 Definition to be joined together Loop: at a single Apoint loopin is space any forclosed their interconnection path atocircuit. be A closed path considered is formed a singleby node. starting An example at a of node, this ispassing shown inthrough Figure 2.3. awhile set of nodes the four elements the figure are connected together, their connection does and returning to the starting node without passing through any node more not occur at a single point in space. Rather, it is a distributed connection. than once. Nodes Loop Elements 2.2 An arbitrary circuit. Branch Definition Series: Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current. Definition Parallel: Two or more elements are in parallel if they are connected to the same two nodes and consequently hae the same oltage across them.

7 DC 2.2. NODE, BRANCHES AND LOOPS Elements may be connected in a way that they are neither in series nor in parallel. Example How many branches and nodes does the circuit in the following figure hae? Identify the elements that are in series and in parallel. 5 Ω 1 Ω 2 Ω 10 V 4 Ω A loop is said to be independent if it contains a branch which is not in any other loop. Independent loops or paths result in independent sets of equations. A network with b branches, n nodes, and l independent loops will satisfy the fundamental theorem of network topology: Elements b = l n Kirchhoff s Laws CHAPTER T Definition The primary signals withinaa circuit are its currents and oltages, which we denote by the symbols i and D, respectiely. We Distributed node define a branch current as the current along a branch of the circuit, and Ideal wires a branch oltage as the potential difference measured across a branch. A D B C B C FIGURE 2.3 interconnectio elements that at a single nod i Branch current - Branch oltage FIGURE 2.4 definitions illu a circuit. 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 oltages, which we denote by the symbols i and, respectiely. We define a branch current as the current along a branch of the circuit (see Figure 2.4), and a branch oltage as the potential difference measured across a branch. Since elements and branches are the same for circuits formed of two-terminal elements, the branch oltages and currents are the same as the corresponding terminal ariables for the elements forming the branches. Recall, as defined in Chapter 1, the terminal ariables for

8 22 2. BASIC LAWS 2.3. Kirchhoff s Laws Ohm s law coupled with Kirchhoff s two laws gies a sufficient, powerful set of tools for analyzing a large ariety of electric circuits Kirchhoff s current law (KCL): the algebraic sum of currents departing a node (or a closed boundary) is zero. Mathematically, i n = 0 n KCL is based on the law of conseration of charge. An alternatie form of KCL is Sum of currents (or charges) drawn as entering a node = Sum of the currents (charges) drawn as leaing the node. i 1 i 5 i 4 i 2 i 3 Note that KCL also applies to a closed boundary. This may be regarded as a generalized case, because a node may be regarded as a closed surface shrunk to a point. In two dimensions, a closed boundary is the same as a closed path. The total current entering the closed surface is equal to the total current leaing the surface. Closed boundary

9 2.3. KIRCHHOFF S LAWS 23 Example A simple application of KCL is combining current sources in parallel. a I T I 1 I 2 I 3 b (a) a I T I T = I 1 I 2 I 3 b (b) A Kirchhoff s oltage law (KVL): the algebraic sum of all oltages around a closed path (or loop) is zero. Mathematically, M m = 0 m=1 KVL is based on the law of conseration of energy. An alternatie form of KVL is Sum of oltage drops = Sum of oltage rises

10 24 2. BASIC LAWS Example When oltage sources are connected in series, KVL can be applied to obtain the total oltage. a V 1 V ab V 2 a V 3 V ab V S = V 1 V 2 V 3 b b Example Find 1 and 2 in the following circuit. 4 Ω 1 10 V 8 V 2 2 Ω Example

11 (a) (b) (c) Fig Question KIRCHHOFF S LAWS 25 6 Res-monster maze. In Fig , all the resistors hae a resistance of 4.0 and all the (ideal) batteries hae an emf of 4.0 Example V. What is the current (HRW through p. 725). resistor In the R? figure (If below, you can all find the the resistors haeproper a resistance loop through of 4.0 this Ω and maze, all you the can (ideal) answer batteries the question are 4.0with V. What a is the current few seconds through of mental resistor calculation.) R? es? (b) Are resistances 2, to a batallel. Rank nt through R 2 x d e uestion 4. hat R 1 R 2 Fig Question 6. 7 A resistor R 1 is wired to a battery, then resistor R 2 is added in series. Are (a) the potential difference across R 1 and (b) the cur-

12 26 2. BASIC LAWS 2.4. Series Resistors and Voltage Diision When two resistors R 1 and R 2 ohms are connected in series, they can be replaced by an equialent resistor R eq where R eq = R 1 R 2. In particular, the two resistors in series shown in the following circuit i a R 1 R b can be replaced by an equialent resistor R eq where R eq = R 1 R 2 as shown below. i a R eq b The two circuits aboe are equialent in the sense that they exhibit the same oltage-current relationships at the terminals a-b. Voltage Diider: If R 1 and R 2 are connected in series with a oltage source olts, the oltage drops across R 1 and R 2 are 1 = R 1 R 1 R 2 and 2 = R 2 R 1 R 2

13 2.5. PARALLEL RESISTORS AND CURRENT DIVISION 27 Note: The source oltage is diided among the resistors in direct proportion to their resistances In general, for N resistors whose alues are R 1, R 2,..., R N ohms connected in series, they can be replaced by an equialent resistor R eq where N R eq = R 1 R 2 R N = If a circuit has N resistors in series with a oltage source, the jth resistor R j has a oltage drop of j = j=1 R j R 1 R 2 R N 2.5. Parallel Resistors and Current Diision When two resistors R 1 and R 2 ohms are connected in parallel, they can be replaced by an equialent resistor R eq where or 1 R eq = 1 R 1 1 R 2 R eq = R 1 R 2 = R 1R 2 R 1 R 2 R j i Node a i 1 i 2 R 1 R 2 Node b Current Diider: If R 1 and R 2 are connected in parallel with a current source i, the current passing through R 1 and R 2 are R 2 R 1 i 1 = i and i 2 = i R 1 R 2 R 1 R 2 Note: The source current i is diided among the resistors in inerse proportion to their resistances.

14 28 2. BASIC LAWS Example Example = Example (a) (na) = Example (ma) (na) = In general, for N resistors connected in parallel, the equialent resistor R eq = R 1 R 2 R N is 1 = R eq R 1 R 2 R N Example Find R eq for the following circuit. 4 Ω 1 Ω R eq 2 Ω 5 Ω 8 Ω 6 Ω 3 Ω Example Find the equialent resistance of the following circuit.

15

16

17

18

19 2.5. PARALLEL RESISTORS AND CURRENT DIVISION 29 i 4 Ω a i 0 12 V 6 Ω 0 3 Ω b Example Find i o, o, p o (power dissipated in the 3Ω resistor). Example Three light bulbs are connected to a 9V battery as shown below. Calculate: (a) the total current supplied by the battery, (b) the current through each bulb, (c) the resistance of each bulb. I I 1 9 V 15 W 10 W 20 W 9 V V 2 V 3 I 2 R 2 R 3 V 1 R 1

20 30 2. BASIC LAWS 2.6. Practical Voltage and Current Sources An ideal oltage source is assumed to supply a constant oltage. This implies that it can supply ery large current een when the load resistance is ery small. Howeer, a practical oltage source can supply only a finite amount of current. To reflect this limitation, we model a practical oltage source as an ideal oltage source connected in series with an internal resistance r s, as follows: Similarly, a practical current source can be modeled as an ideal current source connected in parallel with an internal resistance r s Measuring Deices Ohmmeter: measures the resistance of the element. Important rule: Measure the resistance only when the element is disconnected from circuits. Ammeter: connected in series with an element to measure current flowing through that element. Since an ideal ammeter should not restrict the flow of current, (i.e., cause a oltage drop), an ideal ammeter has zero internal resistance. Voltmeter:connected in parallel with an element to measure oltage across that element. Since an ideal oltmeter should not draw current away from the element, an ideal oltmeter has infinite internal resistance.