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FUNDAMENTALS OF ELECTRICAL ENGINEERING [ ENT 163 ] LECTURE #2 BASIC LAWS HASIMAH ALI Programme of Mechatronics, School of Mechatronics Engineering, UniMAP. Email: [email protected] CONTENTS Introduction Ohm’s Law Nodes, Branches and Loops Kirchhoff’s Laws Series Resistors and Voltage Division Parallel Resistors and current Division Introduction • Fundamental laws – govern electric circuits: 1. Ohm’s Law 2. Kirchhoff’s law • Technique commonly applied: 1. 2. 3. 4. Resistors in series/ parallel Voltage division Current division Delta-to-wye and wye-to-delta transformations Ohm’s Law • Materials in general have a characteristics behavior of resisting the flow of electric charge. • Resistance: ability to resist current (R) . R A Material Resistitivity( m ) Usage Silver 1.64 x 10-8 Conductor Copper 1.72 x 10-8 Conductor Aluminum 2.8 x 10-8 Conductor Gold 2.45 x 10-8 Conductor Carbon 4 x 10-5 Semiconductor Germanium 47 x 10-2 Semiconductor Silicon 6.4 x 102 Semiconductor Paper 1010 Insulator Mia 5 x 1011 Insulator Glass 1012 Insulator Teflon 3 x 1012 Insulator Ohm’s Law • Resistor: circuit element that used to model the current – resisting behavior. •Relationship between current and voltage for a resistor Ohm’s law Ohm’s law states that the voltage v across a resistor is directly proportionally to the current i flowing through the resistor. • Georg Simon Ohm (1787 – 1854), a German physicist, is credited with finding the relationship between current and voltage for a resistor. • Ohm- defined the constant of proportionality for a resistor resistance, R v iR • Resistance, R: denotes its ability to resist the flow of electric current(Ω) • The two extreme possible values of R: • R= 0 is called a short circuit (v= i R=0) • R=∞ is called open circuit v i lim R R 0 Ohm’s Law A short circuit is a circuit element with resistance approaching zero An open circuit is a circuit element with resistance approaching infinity. • Resistor is either fixed or variable. • Fixed resistors have constant resistor. • The two common types of fixed resistors: • Wire wound • Composition • Variable resistor have adjustable resistance. Example potentiometer. • Reciprocal of resistance, known as conductance, G: G 1 i R v The conductance is the ability of an element to conduct electric current , it is measured in mhos or Siemens (S). Ohm’s Law • The power is dissipated by a resistor is expressed by: v2 p vi i R R 2 or 2 i p vi v 2G G Note: 1. The power dissipated in a resistor is a nonlinear function of either current or voltage. 2. Since R and G are positive quantities, the power dissipated in a resistor is always positive( resistor- absorbs power from circuit. Thus resistor is a passive element, incapable of generating energy. Nodes, Branches and Loops • A network – interconnection of elements or devices. • A circuit – network providing one/ more closed paths. • Branch represents a single elements such as a voltage source or resistor. A branch represents any two-terminal element. • Node is the point of connection between two or more branches. • It usually indicated by a dot in a circuit. • Loop is any closed path in a circuit ; formed by starting at a node, passing through a set of nodes, and returning to the starting node without passing through ant node more than once. 5Ω b a 10 V + - 2Ω 3Ω c 2A Nodes, Branches and Loops 1. Two or more elements are in series if they exclusively share a single node and consequently carry the same current. 2. Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them. Nodes, Branches and Loops Example: Determine the number of branches and nodes in the circuit shown in Figure 2.12. Identify which elements are in series and which are in parallel. 5Ω 10 V + - 6Ω Figure 2.12 2A Kirchhoff’s Law • First introduced in 1874 by the German physicist Gustav Robert Kirchhoff (1824-1887) • Kirchhoff’s current law (KCL): 1. The algebraic sum of currents entering a node (or close boundary) is zero. 2. The sum of the currents entering a node is equal to the sum of the currents leaving the node. N i n 1 n 0 i1 ()i2 i3 i4 ()i5 0 i5 i1 i3 i2 i4 Fig: Currents at a node illustrating KCL i1 i3 i4 i2 i5 Kirchhoff’s Law IT IT a a I1 I2 I3 I S I1 I 2 I 3 = b b IT I 2 I1 I 3 IT I1 I 2 I 3 Kirchhoff’s Law • Kirchhoff’s voltage law (KVL): 1. The algebraic sum of all voltages around a closed path (or loop) is zero. M v m 1 m 0 2. Sum of voltage drops is equal to sum of voltage rises. + v2 - v1 + v3 - + - v1 v2 v3 v4 v5 0 + v4 v2 v3 v5 v1 v4 - v5 + Sum of voltage drop = Sum of voltage rises Kirchhoff’s Law • Kirchhoff’s voltage law (KVL): a + Vab + V1 + V2 a+ = b + b - + Vab VS V1 V2 V3 - V3 v2 v3 v5 v1 v4 Kirchhoff’s Law Example: For the circuit in Fig. 2.21 find voltages v1 and v2. 2Ω + v1 20V + - v2 + 3Ω Kirchhoff’s Law Problem: Find v1and v2 in the circuit of the Fig. 4Ω + v1 10V + - 8V + v2 2Ω + Kirchhoff’s Law Problem: Determine voand i in the circuit of the Fig. i 4Ω 2vo + - + v1 12V + - 4V + vo - 6Ω + Kirchhoff’s Law Problem: Find current io and voltage vo in the circuit shown in Fig. below. a io + vo - 0.5io 4Ω 3A Series Resistors and Voltage Division Consider a single-loop circuit with two resistors in series i v •a + v1 - + v2 - R1 R2 Applying Ohm’s law of each resistor v1 iR + - (1) If apply KVL to the loop (CW), we have b i •a + - + v Req - Combing (1) and (2): v v1 v2 i( R1 R2 ) v iReq b • (2) v v1 v2 0 • v v2 iR2 or i Req R1 R2 v R1 R2 Series Resistors and Voltage Division The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances. Mathematically, N Req R1 R2 R3 ... RN RN n 1 To determine the voltage across each resistor shown in Fig, v1 R1 v R1 R2 , v2 Principle of voltage division: vn RN v R1 R2 ... RN R2 v R1 R2 Parallel Resistors And Current Division • The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum. • Mathematically, Req R1 R2 R1 R2 • For a circuit with N resistors in parallel, 1 1 1 1 .. Req R1 R2 RN Parallel Resistors And Current Division Consider the circuit below where two resistors are connected in parallel and therefore have the same voltage across them. From Ohm’s law Node a i i2 i1 v + - R2 R1 i1 Node b Substitute (3) into (4), we get i v v 1 1 v v( ) , R1 R2 R1 R2 Req 1 1 1 Req R1 R2 v1 i1R1 i2 R2 Req R1 R2 R1 R2 v , R1 i2 v , R2 (3) Applying KCL at node a gives i i1 i2 (4), • For a circuit with N resistors in parallel, 1 1 1 1 .. Req R1 R2 RN Parallel Resistors And Current Division • Principle of current division: i1 • R2i R1 R2 i2 R1i R1 R2 Extreme cases: 1. R2=0,-Req=0 ; entire current flows through the short circuit. 2. R2= ∞ , Req=R1; current flows through the path of least resistance. Wye – Delta Transformation •Implementation – three – phase networks, electrical filters, etc. •Delta to wye conversion: each resistor in the Y network is the product of the resistors in the two adjacent ∆ branches, divided by the sum of the three ∆ resistors. R1 Rb Rc Ra Rb Rc R2 Rc Ra Ra Rb Rc R3 Ra Rb Ra Rb Rc Wye – Delta Transformations Wye to delta conversion: each resistor in the network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor. Ra R1 R2 R2 R3 R3 R1 R1 Rb R1 R2 R2 R3 R3 R1 R2 Rc R1 R2 R2 R3 R3 R1 R3 Wye – Delta Transformations Y and ∆ networks are said to be balanced when R1 R2 R3 RY Ra Rb Rc R Therefore conversion formulas: R1 R 3 R 3RY