1 Sirindhorn International Institute of Technology Thammasat University School of Information, Computer and Communication Technology Lecture Notes: ECS 203 Basic Electrical Engineering Semester 1/2010 Dr.Prapun Suksompong 1 June 16, Special thanks to Dr.Waree Kongprawechnon and Dr.Somsak Kittipiyakul for earlier versions of the notes. Parts of the notes were compiled from C.K. Alexander and M.N.O. Sadiku, Fundamentals of Electric Circuits, 4th ed., McGraw-Hill, International Edition, 2009 and G. Rizzoni, Principles and Applications of Electrical Engineering, 5th ed., Mc-Graw-Hill, International Edition, 2007
2 CHAPTER 1 Basic Concepts In electrical engineering, we are often interested in communicating or transferring energy from one point to another. To do this requires an interconnection of electrical devices. Such interconnection is referred to as an electric circuit, and each component of the circuit is known as an element. Definition An electric circuit is an interconnection of electrical elements Systems of Units As engineers, we deal with measurable quantities. Our measurement must be communicated in standard language that virtually all professionals can understand irrespective of the country. Such an international measurement language is the International System of Units (SI). In this system, there are six principal units from which the units of all other physical quantities can be derived. 7
3 8 1. BASIC CONCEPTS Quantity Basic Unit Symbol Length meter m Mass kilogram Kg Time second s Electric Current ampere A Temperature kelvin K Luminous Intensity candela cd Charge coulomb C One great advantage of SI unit is that it uses prefixes based on the power of 10 to relate larger and smaller units to the basic unit. Example Change of units: Multiplier Prefix Symbol tera T 10 9 giga G 10 6 mega M 10 3 kilo k 10 2 centi c 10 3 milli m 10 6 micro µ 10 9 nano n pico p 600, 000, 000 mm = 1.2. Circuit Variables Charge: The concept of electric charge is the underlying principle for all electrical phenomena. Charge is an electrical property of the atomic particles of which matter consists, measured in coulombs (C). The charge of an electron is C. The coulomb is a large unit for charges. In 1 C of charge, there are 1/( ) = electrons. Thus realistic or laboratory values of charges are on the order of pc, nc, or µc. A large power supply capacitor can store up to 0.5 C of charge Law of Conservation of Charge: Charge can neither be created nor destroyed, only transferred.
4 1.2. CIRCUIT VARIABLES 9 Definition Current: The time rate of change of charge, measured in amperes (A). Mathematically, i(t) = d dt q(t) Note: 1 ampere (A) = 1 coulomb/second (C/s). The charge transferred between time t 1 and t 2 is obtained by q = t2 t 1 idt Two types of currents: (a) A direct current (DC) is a current that remains constant with time. (b) An alternating current (AC) is a current that varies with time. Such AC current is used in your household, to run the air conditioner, refrigerator, washing machine, and other electric appliances.
5 10 1. BASIC CONCEPTS By convention the symbol I is used to represent such a constant current. A time-varying current is represented by the symbol i. Definition Voltage (or potential difference): the energy required to move a unit charge though an element, measured in volts (V). The voltage between two points a and b in a circuit is denoted by v ab and can be interpreted in two ways: (a) point a is at a potential of v ab volts higher than point b, or (b) the potential at point a with respect to point b is v ab. Note: 1 volt (V) = 1 joule/coulomb = 1 newton-meter/coulomb v ab = v ba Mathematically, v ab = dw dq where w is the energy in joules (J) and q is charge in coulombs (C). The plus (+) and minus (-) signs at the points a and b are used to define reference direction or voltage polarity.
6 1.2. CIRCUIT VARIABLES Like electric current, a constant voltage is called a DC voltage and is represented by V, whereas a sinusoidally time-varying voltage is called an AC voltage and is represented by v. A dc voltage is commonly produced by a battery; ac voltage is produced by an electric generator Current and voltage are the two basic variables in electric circuits. The common term signal is used for an electric quantity such as a current or a voltage (or even electromagnetic wave) when it is used for conveying information. Engineers prefer to call such variables signals rather than mathematical functions of time because of their importance in communications and other disciplines. For practical purposes, we need to be able to find/calculate/measure more than the current and voltage. We all know from experience that a 100-watt bulb gives more light than a 60-watt bulb. We also know that when we pay our bills to the electric utility companies, we are paying for the electric energy consumed over a certain period of time. Thus power and energy calculations are important in circuit analysis. Definition Power: time rate of expending or absorbing energy, measured in watts (W). Mathematically, the instantaneous power p = dw dt = dw dq dq dt = vi Definition Sign of power Plus sign: Power is absorbed by the element. (resistor, inductor) Minus sign: Power is supplied by the element. (battery, generator) Definition Passive sign convention: If the current enters through the positive polarity of the voltage, p = vi. If the current enters through the negative polarity of the voltage, p = vi.
7 12 1. BASIC CONCEPTS Law of Conservation of Energy: Energy can neither be created nor destroyed, only transferred.
8 1.2. CIRCUIT VARIABLES 13 For this reason, the algebraic sum of power in a circuit, at any instant of time, must be zero. The total power supplied to the circuit must balance the total power absorbed. Example Energy: the energy absorbed or supplied by an element from time 0 to t is w = t 0 p dt = t 0 vi dt. Integration suggests finding area under the curve. Need to be careful with negative area. The electric power utility companies measure energy in kilowatt-hours (kwh), where 1 kwh = 3600 kj.
9 14 1. BASIC CONCEPTS 1.3. Circuit Elements Definition There are 2 types of elements found in electrical circuits. 1) Active elements (is capable of generating energy), e.g., generators, batteries, and operational amplifiers (Op-amp). 2) Passive element, e.g., resistors, capacitors and inductors The most important active elements are voltage and current sources: (a) Voltage source provides the circuit with a specified voltage (e.g. a 1.5V battery) (b) Current source provides the circuit with a specified current (e.g. a 1A current source). Definition In addition, we may characterize the voltage or current sources as: 1) Independent source: An active element that provides a specified voltage or current that is completely independent of other circuit elements.
10 1.3. CIRCUIT ELEMENTS 15 2) Dependent source: An active element in which the source quantity is controlled by another voltage or current The key idea to keep in mind is that a voltage source comes with polarities (+ -) in its symbol, while a current source comes with an arrow, irrespective of what it depends on Remarks: Dependent sources are useful in modeling elements such as transistors, operational amplifiers and integrated circuits. An ideal voltage source (dependent or independent) will produce any current required to ensure that the terminal voltage is as stated. An ideal current source will produce the necessary voltage to ensure the stated current flow. Thus an ideal source could in theory supply an infinite amount of energy. Not only do sources supply power to a circuit, they can absorb power from a circuit too. For a voltage source, we know the voltage but not the current supplied or drawn by it. By the same token, we know the current supplied by a current source but not the voltage across it.
11 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 Ohm s Law Ohm s law shows a relationship between voltage and current of a resistive element such as conducting wire or light bulb Ohm s Law: The voltage v across a resistor is directly proportional 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 implies that current flows from a higher potential to a lower potential 17
12 18 2. BASIC LAWS in order for v = ir. If current flows from a lower potential to a higher potential, v = ir The resistance R of a cylindrical conductor of cross-sectional area A, length L, and conductivity σ is given by R = L σa. Alternatively, R = ρ L 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 Remarks: (a) R = v/i (b) Conductance : G = 1 R = i v The unit of G is the mho 1 ( ) or siemens 2 (S) 1 Yes, this is NOT a typo! It was derived from spelling ohm backwards. 2 In English, the term siemens is used both for the singular and plural.
13 2.1. OHM S LAW 19 (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. (ii) When R =, we have an open circuit and v i = lim R R = 0 indicating that i = 0 for any v A resistor is either fixed or variable. Most resistors are of the fixed type, meaning their resistance remains constant.
14 20 2. BASIC LAWS A common variable 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 devices with nonlinear resistance are the lightbulb and the diode. Although all practical resistors may exhibit nonlinear behavior under certain conditions, we will assume in this class that all elements actually designated as resistors are linear.
15 2.1. OHM S LAW Using Ohm s law, the power p dissipated by a resistor R is p = vi = i 2 R = v2 R. Example In the circuit below, calculate the current i, and the power p. Definition 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 Determine the minimum resistor size that can be connected to a 1.5V battery without exceeding the resistor s 1 4-W power rating.
16 22 2. BASIC LAWS 2.2. Node, Branches and Loops Definition Since the elements of an electric circuit can be interconnected in several ways, we need to understand some basic concept of network topology. Network = interconnection of elements or devices Circuit = a network with closed paths Definition Branch: A branch represents a single element such as a voltage source or a resistor. A branch represents any two-terminal element. Definition Node: A node is the point of connection between two or more branches. It is usually indicated by a dot in a circuit. If a short circuit (a connecting wire) connects two nodes, the two nodes constitute a single node. Definition Loop: A loop is any closed path in a circuit. A closed path is formed by starting at a node, passing through a set of nodes and returning to the starting node without passing through any node more than once. 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 have the same voltage across them. 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 have? Identify the elements that are in series and in parallel.
17 2.3. KIRCHHOFF S LAWS 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: b = l + n Kirchhoff s Laws Ohm s law coupled with Kirchhoff s two laws gives a sufficient, powerful set of tools for analyzing a large variety of electric circuits Kirchhoff s current law (KCL): the algebraic sum of current entering a node (or a closed boundary) is zero. Mathematically, N i n = 0 n=1 KCL is based on the law of conservation of charge. An alternative form of KCL is Sum of currents (or charges) entering a node = Sum of the currents (charges) leaving the node. 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 leaving the surface.
18 24 2. BASIC LAWS Example A simple application of KCL is combining current sources in parallel. A Kirchhoff s voltage law (KVL): the algebraic sum of all voltages around a closed path (or loop) is zero. Mathematically, M v m = 0 m=1 KVL is based on the law of conservation of energy. An alternative form of KVL is Sum of voltage drops = Sum of voltage rises.
19 2.3. KIRCHHOFF S LAWS 25 Example When voltage sources are connected in series, KVL can be applied to obtain the total voltage. Example Find v 1 and v 2 in the following circuit.
20 26 2. BASIC LAWS 2.4. Series Resistors and Voltage Division When two resistors R 1 and R 2 ohms are connected in series, they can be replaced by an equivalent resistor R eq where R eq = R 1 + R 2. In particular, the two resistors in series shown in the following circuit can be replaced by an equivalent resistor R eq where R eq = R 1 + R 2 as shown below. The two circuits above are equivalent in the sense that they exhibit the same voltage-current relationships at the terminals a-b. Voltage Divider: If R 1 and R 2 are connected in series with a voltage source v volts, the voltage drops across R 1 and R 2 are v 1 = R 1 R 1 + R 2 v and v 2 = R 2 R 1 + R 2 v
21 2.5. PARALLEL RESISTORS AND CURRENT DIVISION 27 Note: The source voltage v is divided among the resistors in direct proportion to their resistances In general, for N resistors whose values are R 1, R 2,..., R N ohms connected in series, they can be replaced by an equivalent resistor R eq where N R eq = R 1 + R R N = If a circuit has N resistors in series with a voltage source v, the jth resistor R j has a voltage drop of v j = j=1 R j R 1 + R R N v 2.5. Parallel Resistors and Current Division When two resistors R 1 and R 2 ohms are connected in parallel, they can be replaced by an equivalent resistor R eq where or 1 R eq = 1 R R 2 R eq = R 1 R 2 = R 1R 2 R 1 + R 2 R j Current Divider: 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 divided among the resistors in inverse proportion to their resistances.
22 28 2. BASIC LAWS Example Example = Example (a) (na) = Example (ma) (na) = In general, for N resistors connected in parallel, the equivalent 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. Example Find i o, v 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.
23 2.7. MEASURING DEVICES Practical Voltage and Current Sources An ideal voltage source is assumed to supply a constant voltage. This implies that it can supply very large current even when the load resistance is very small. However, a practical voltage source can supply only a finite amount of current. To reflect this limitation, we model a practical voltage source as an ideal voltage 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 Devices 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 voltage drop), an ideal ammeter has zero internal resistance. Voltmeter:connected in parallel with an element to measure voltage across that element. Since an ideal voltmeter should not draw current away from the element, an ideal voltmeter has infinite internal resistance.
24 CHAPTER 3 Methods of Analysis Here we apply the fundamental laws of circuit theory (Ohm s Law & Kirchhoff s Laws) to develop two powerful techniques for circuit analysis. 1. Nodal Analysis (based on KCL) 2. Mesh Analysis (based on KVL) This is the most important chapter for our course Nodal Analysis Analyzing circuit using node voltages as the circuit variables Steps to Determine Node Voltages: Step 0: Determine the number of nodes n. Step 1: Select a node as a reference node (ground node). Assign voltages v 1, v 2,, v n 1 to the remaining n 1 nodes. The voltage are referenced with respect to the reference node. The ground node is assumed to have 0 potential. Step 2: Apply KCL to each of the n 1 nonreference nodes. Use Ohm s law to express the branch currents in terms of node voltages. Step 3: Solve the resulting simultaneous equations to obtain the unknown node voltages Remark: 31
25 32 3. METHODS OF ANALYSIS (a) Current flows from a higher potential to a lower potential in a resistor. (b) If a voltage source is connected between the reference node and a nonreference node, we simply set the voltage at the nonreference node equal to the voltage of the voltage source. (c) Multiple methods to solve the simultaneous equations in Step 3. Method 1: Elimination technique (good for a few variables) Method 2: Write in term of matrix and vectors (write Ax = b), then use Cramer s rule. Method 3: Use computer (e.g., Matlab) to find A 1 and x = A 1 b Example Calculate the node voltages in the circuit below.
26 3.1. NODAL ANALYSIS 33 Example Calculate the node voltages in the circuit below Special Case: If there is a voltage source connected between two nonreference nodes, the two nonreference nodes form a supernode. We apply both KCL and KVL to determine the node voltages. Example Find v and i in the circuit below.
27 34 3. METHODS OF ANALYSIS Remarks: Note the following properties of a supernode: (a) The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages. (b) A supernode has no voltage of its own. (c) We can have more than two nodes forming a single supernode. (d) The supernodes are treated differently because nodal analysis requires knowing the current through each element. However, there is no way of knowing the current through a voltage source in advance Mesh Analysis Mesh analysis provides another general procedure for analyzing circuits, using mesh currents as the circuit variables. Definition Mesh is a loop which does not contain any other loop within it Steps to Determine Mesh Currents: Step 0: Determine the number of meshes n. Step 1: Assign mesh current i 1, i 2,, i n, to the n meshes. The direction of the mesh current is arbitrary (clockwise or counterclockwise) and does not affect the validity of the solution. For convenience, we define currents flow in the clockwise (CW) direction. Step 2: From the current direction in each mesh, define the voltage drop polarities. Step 3: Apply KVL to each of the n meshes. Use Ohm s law to express the voltages in terms of the mesh current. Step 4: Solve the resulting n simultaneous equations to get the mesh current.
28 3.2. MESH ANALYSIS 35 Example Find the branch currents I 1, I 2, and I 3 using mesh analysis.
29 36 3. METHODS OF ANALYSIS Remarks: (a) Nodal analysis applies KCL to find unknown voltages in a given circuit, while mesh analysis applies KVL to find unknown currents. (b) Using mesh currents instead of element currents as circuit variables is convenient and reduces the number of equations that must be solved simultaneously. (c) Mesh analysis is not quite as general as nodal analysis because it is only applicable to a circuit that is planar. A planar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it is nonplanar Nodal Versus Mesh Analysis You should be familiar with both methods. However, given a network to be analyzed, how do we know which method is better or more efficient? Suggestion: Choose the method that results in smaller number of variables or equations. A circuit with fewer nodes than meshes is better analyzed using nodal analysis, while a circuit with fewer meshes than nodes is better analyzed using mesh analysis. You can also use one method to check your results of the other method.
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