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1. INTRODUCTION Fundamental laws – govern electric circuits: a) Ohm’s law b) Kirchhoff’s law Techniques commonly applied: a) resistors in series / parallel b) voltage division c) current division d) delta-to-wye and wye-to-delta transformations 2. OHM’S LAW Materials – have a characteristic behavior of resisting the flow of electric charge Resistance: ability to resist current (R) Mathematically, R A Fig. 2.1: a) Resistor b) Circuit symbol for resistance Material Silver Copper Aluminum Gold Carbon Germanium Silicon Paper Mica Glass Teflon Resistivity (.m) 1.64 x 10-8 1.72 x 10-8 2.8 x 10-8 2.45 x 10-8 4 x 10-5 47 x 10-2 6.4 x 102 1010 5 x 1011 1012 3 x 1012 Usage Conductor Conductor Conductor Conductor Semiconductor Semiconductor Semiconductor Insulator Insulator Insulator Insulator Resistivities of common materials Resistor: circuit element – used to model the current-resisting behavior Relationship between current and voltage for a resistor – Ohm’s Law Ohm’s Law: the voltage v across a resistor is directly proportional 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 Resistance, R: denotes its ability to resist the flow of electric current () Two extreme possible values of R: a) R = 0 b) R = short circuit (v = i R = 0) lim v 0 open circuit i R R Short circuit: circuit element with resistance approaching zero Open circuit: circuit element with resistance approaching infinity Fig. 2.2: a) Short circuit b) Open Circuit Fixed Resistor Variable Fig. 2.3: Symbol for a) variable resistor, b) potentiometer Conductance, G – how well an element will conduct electric current G 1 i R v Note: Conductance: the ability of an element to conduct electric current; measured in mho or siemens (S) Power dissipated by a resistor: v2 p vi i R R 2 OR p vi v 2 G i2 G 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, resistor – passive element, incapable in generating energy) 3. NODES, BRANCHES AND LOOPS Network – interconnection of elements or devices Circuit – network providing one / more closed paths Branch: represents a single element (e.g. voltage source, resistor); i.e. represents any two-terminal element Node: point of connection between two or more branches; usually indicated by a dot in a circuit Loop: 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 any node more than once. Note: 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. Fig. 2.4: Nodes, branches and loops 4. KIRCHHOFF’S LAWS First introduced in 1874 by the German physicist Gustav Robert Kirchhoff (1824-1887) Kirchhoff’s current law (KCL): a) The algebraic sum of currents entering a node (or closed boundary) is zero. b) The sum of the currents entering a node is equal to the sum of the currents leaving the node. c) Mathematically, N i n 1 n 0 Kirchhoff’s voltage law (KVL): a) The algebraic sum of all voltages around a closed path (or loop) is zero. b) Sum of voltage drops = sum of voltage rises c) Mathematically, M v m 1 Fig. 2.5: KCL m 0 Fig. 2.6: KVL 5. 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 R N Rn n 1 Principle of voltage division: vn Rn v R1 R2 RN Fig. 2.7: a) A single-loop circuit with two resistors in series, b) equivalent circuit 6. 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, RR Req 1 2 R1 R2 For a circuit with N resistors in parallel, 1 1 1 1 Req R1 R2 RN Principle of current division: i1 R2 i R1 i , i2 R1 R2 R1 R2 Extreme cases: a) R2 = 0, Req = 0 (refer equation); entire current flows through the short circuit b) R2 = , Req = R1; current flows through the path of least resistance Fig. 2.8: a) Two resistors in parallel, b) equivalent circuit Fig. 2.9: a) A shorted circuit, b) an open circuit 7. WYE – DELTA TRANSFORMATIONS 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 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. R R R2 R3 R3 R1 Ra 1 2 R1 Rb R1 R2 R2 R3 R3 R1 R2 Rc R1 R2 R2 R3 R3 R1 R3 Y and networks are said to be balanced when R1 = R2 = R3 = RY, R Ra = Rb = Rc = Therefore conversion formulas: RY R 3 or R 3 RY Fig. 2.10: Wye-delta transformations