# Resistor. l A. Factors affecting the resistance are 1. Cross-sectional area, A 2. Length, l 3. Resistivity, ρ

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1 Chapter 2 Basic Laws. Ohm s Law 2. Branches, loops and nodes definition 3. Kirchhoff s Law 4. Series resistors circuit and voltage division. 5. Equivalent parallel circuit and current division. 6. Wye-Delta transformations

2 Learning Outcome... At the end of this topic, students should be able to: Explain and use Ohm s Law State the Kirchhoff s Current and Voltage Law Solve the circuit s problems by utilising the Ohm s Law, Kirchhoff s Current and Voltage Law, etc.

3 esistor When a current is flow through a material, some of free electrons experiences collisions with atoms restricting the movement due to energy loss restriction varies with the material type We called this property as resistance Factors affecting the resistance are. Cross-sectional area, A 2. Length, l 3. esistivity, ρ = ρ l A

4 esistor (cont.) esistance () is the opposition to current. Unit: ohms, Ω Conductance (G) is the reciprocal of resistance. Unit: Siemens, S

5 Cont Good conductors copper and aluminum [low resistivities] Insulators mica and paper [high resistivities]

6 esistor (cont.) esistor is a component designed to have a certain amount of resistance. Application To limit current in the circuit, To divide the voltage Two categories: ) Fixed resistors (resistance value is fixed during manufacturing) 2) Variable resistors (resistance value can be changed) Potentiometer Potentiometer connected as rheostat heostat

7 Colour-band 5-band colour code Digit 0 Colour Black esistance value, first three bands: 2 Brown ed st band st digit 3 Orange 4-band colour code esistance value, first three bands: st band st digit 2 nd band 2 nd digit 3 rd band multiplier (i.e. number of zeros following the 2 nd digit) Fourth band - tolerance Digit ±5% ±0% Colour Black Brown ed Orange Yellow Green Blue Violet Gray White Gold Silver 2 nd band 2 nd digit 3 rd band 3 rd digit 4 th band - multiplier (i.e. number of zeros following the 3 rd digit) Fourth band multiplier Fifth band tolerance ±2% ±% ±0.5% ±0.25% ±0.% Yellow Green Blue Violet Gray White Gold Silver ed Brown Green Blue Violet

8 2. Ohm s Law Ohm s law states that current is directly proportional to voltage and inversely proportional to resistance. (source: Floyd) V I = It could also be defined as for an ideal resistor, the instantaneous value of the voltage is proportional to the instantaneous value of the current. (source: Mersereau) ( t) i( t) v =

9 Ohm s Law (cont.) To apply v = i, the current and voltage must conform with the passive sign convention. This implies that the current flows from a higher potential to a lower potential (v=i). If current flows from a lower potential to high potential, v=-i i

10 Example Determine the value of current, i if the voltage, v is 0 V and is 0 kω (Fig. ). How much power is dissipated by resistor,? Fig. eminder: Please take note on the direction of current flow. Always remember to draw the arrow to indicate its direction.

11 Short Circuit If = 0, short circuit In practical, short circuit perfect conductor resistance approaching zero v = i = 0 V, voltage is zero

12 Open Circuit If =, open circuit Open circuit circuit element with resistance approaching infinity i = lim v = 0 Current is zero, as there is no complete current path

13 2.2 Branches, loops and nodes definition Branch: one current path in a parallel circuit; a current path that connects two nodes Voltage across parallel branches is the same Loop: a closed current path in a circuit Node: a point in a circuit where two or more components are connected (aka a junction) Node Branch

14 4 branches

15 3 nodes

16 3 loops A loop: any closed path in circuit

17 Exercise How many loops are there in this network?

18 2.3 (a) Kirchhoff s Voltage Law Kirchhoff s Voltage Law (KVL) states that the sum of the element and source voltages around any closed path in a circuit is equal to zero. N n= V n = 0 Where, V n = the nth voltage N= number or voltages in the loop

19 Cont To use KVL, assign an algebraic sign (reference direction) to each voltage in the loop. I v v v v = Note: Follow the direction of current flow, I 0

20 Example Write the KVL equation for the following circuit.

21 Example 2 Determine the unknown voltage drop, V 3 in Figure below if V = V 2 = 2 V 3. Figure

22 Example 3 For the following circuit, find voltages v and v 2. 2 Ω 20 V i v v 2 3 Ω

23 Solution 20 v v2 = 0 20 = v v2 () (KVL) v = 2i (2) (Ohm s Law) v2 = 3(-i) = -3i (3) Substituting Eq.(2) and (3) into Eq.() yields; 20 = 2i (-3i) = 5i i = 4A Substituting i = 4A into Eq.(2) and (3), v = 2(4) = 8V. v2 = -3(4) = -2V.

24 Example 4 Determine v o and i in the following figure. i 2 Ω 2v o 20 V i 4 V v o 2 Ω

25 2.3 (b) Kirchhoff s Current Law Kirchhoff s Current Law (KCL) states that the sum of the currents entering a node is equal to the sum of the currents leaving a node. I in I out5 I in2 I in3 I in4 I in5 Current entering the node I out4 I out3 I out2 I out Current leaving the node I I I I I = I I I I in in2 in3 in4 in5 out out 2 out3 out 4 out5 I Another definition: The sum of all the currents entering a node is equal to zero. I ( I ) ( I ) ( I ) ( I ) ( I ) 0 in Iin2 Iin3 Iin4 Iin5 out out2 out3 out 4 out5 =

26 Example Determine the current, I 2 if the currents I T, I and I 3 are 00 ma, 30 ma and 20 ma respectively.

27 2.4 Series resistors circuit and voltage division. A series circuit provides only one path for current to flow the same current flows through each resistor. total resistor values add total = (e.g. above) A circuit consisting of a series string of resistors connected to a voltage source acts as a voltage divider.

28 2.4 Series resistors circuit and voltage division (cont.) Voltage divider states that the voltage drop across any resistor or combination of resistors in a series circuit is equal to the ratio of that resistance value to the total resistance, multiplied by the source voltage. x V x = t V cc

29 Example Figure shows a series circuit. Given that V CC = 5 V, = 3 = 0 kω and 2 = 5 kω. Determine i. total resistance ii. the voltage across 3 iii. the voltage across point A and B iv. the current flow in this series circuit. Figure

30 2.5 Equivalent parallel circuit and current division. The resistors are said to be in parallel with each other when they are connected between two separate nodes in the circuit. the voltage across each branch is the same total resistor, total total = G total = 2 3

31 2.5 Equivalent parallel circuit and current division (cont.) A parallel circuit acts as a current divider because the current entering the junction of parallel branches divides up into several individual branch currents. Current divider states that the current I X through any branch equals the total parallel resistance, T divided by the resistance X of the branch and multiplied by the total current I T into the junction of parallel branches. T I X = X I T

32 Example For the following circuit, find; (a) Equivalent resistance (b) Total current (c) Currents through each resistor (d) Power for each element and total circuit power (e) Power supplied by the source

33 Example 2 Figure shows a parallel circuit. Given that the value of = 2 = 0 kω and 3 = 5 kω. If I CC is 0. ma, determine i. the current through each resistor, i.e. I, I 2 and I 3. ii. the total equivalent resistance. iii. the voltage across point A to B, V AB.

34 Exercise Figure shows a parallel circuit. Given that the value of = 2 = 0 kω, 3 = 5 kω and 4 = 20 kω. If I CC = ma, determine i. the current through each resistor, i.e. I, I 2, I 3 and I 4. ii. the voltage across point A to B, V AB. iii. the voltage across each resistor.

35 Special case: Two-parallel resistors Total resistance of two parallel resistors are: Current divider: T T = = = T T CC T I I I I = = = I T I = 2 2

36 Example Given that I CC = ma, the value of = 0 kω, 2 = 5 kω and 3 = 20 kω. Determine i. the currents I T, I and I 2. ii. the voltage across point A to B, V AB.

37 2.6 Wye-Delta transformations Conversions between Y- and vice-versa are very useful in reducing the complexity of a circuit during analysis process. Take note on the letter subscripts used to designate resistors in the circuit and that numerical subscripts used to designate the resistors in the Y circuit.

38 to Y conversion The conversion rule is as follows: Each resistor in the Y is equal to the product of the resistors in two adjacent branches, divided by the sum of all three resistors. C B A B A C B A C B C B A C A = = = 3 2

39 Y to conversion The conversion rule is as follows: Each resistor in the is equal to the sum of all the possible products of the Y resistors taken two at a time, divided by the opposite Y resistors C B A = = =

40 Example Find its equivalent circuit.

41 Example Find its equivalent Y circuit.

42 Example Obtain the equivalent resistance ab of the circuit shown below.

43 Exercise Given that I CC = ma, the value of = 0 kω, 2 = 5 = 5 kω, 3 = 4 = 20 kω and 6 = 5 kω. Find the voltage across the resistor.

44 Assignment Derive the formula for delta-wye transformation.

45 eferences Thomas L. Floyd, Principles of Electric Circuits: Conventional Current Version, 9 th edition, Pearson, 200 ussell M. Mersereau and Joel. Jackson, Circuit Analysis: A System Approach, Pearson-Prentice Hall, 2006 obert L. Boylestad, Introductory Circuit Analysis, 2 th edition, Pearson, 200

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