Chapter 2 Shared.pptx hjduoihdsauiodsahuidsahusdiahui

ENA electrical network analysis

1. E E-110 Electric Circuits Yumna Bilal D epar t m ent of El e c t r i c al E ng i neer in g University of Gujrat [email protected] 1

2. R es i s t anc e Different material allow charges to move within them with different levels of ease. This physical property or ability to resist current is known as resistance. The resistance of any material with a uniform cross-sectional area A and length l is inversely proportional to A and directly proportional to l . 3

3. R es i s t anc e The constant material, i.e.,  of t he p r o po rt ionali t y is t he r e s i s t i v i t y of t he A R  l A l R   11

4. R es i s t anc e I n ho n or of Ge or ge S i m on O h m ( 178 7 - 1 85 4 ), a G e r m an physicist, the unit of resistance is named Ohm (  ). A conductor designed to have a specific resistance is called a resistor. 12

5. Ohm’s Law The voltage v across a resistor is directly proportional to the c ur rent i f lo w ing th ro ug h t he r e s i s t or. The p ro po rt ionali t y constant is the resistance of the resistor, i.e., v ( t )  Ri ( t ) One can also write: Instantaneous power dissipated in a resistor R i ( t )  1 v ( t )  i ( t )  Gv ( t ) 2 R v 2 ( t ) p ( t )  v ( t ) i ( t )   R i ( t ) 13

6. Linear and Nonlinear Resistors Linear resistor Nonlinear resistor I n t his c o u r s e, w e a s su m e t hat all t he ele m e n t s t h a t ar e designated as resistors are linear (unless mentioned otherwise) 7

7. Resistors (Fixed and Variable) Fixed resistors have a resistance that remains constants. Two common type of fixed resistors are: wirewound composition (carbon film type) 8

8. Fixed Resistors Inside the resistor A common type of resistor that you will work with in your labs: It has 4 color-coded bands (3 for value and one for tolerance) – How to read the value of the resistor? 9

9. Variable Resistors Variable resistors have adjustable resistance and are typically called potentiometer (or pot for short). Potentiometers have three terminals one of which is a sliding contact or wiper. 10

10. C onduc t anc e G = 1 / R is c alled t he co nd u c t a n c e o f th e el e m e nt a nd is  measured in siemens (S) or mho ( ) . German inventor Ernst Werner von Siemens (1816-1892) Conductance is the ability of an element to conduct current.. 11 A device with zero (no) resistance has infinite conductance and a device with infinite resistance has zero conductance.

11. Short and Open Circuits A device with zero resistance is called short circuit and a device with zero conductance (i.e., infinite resistance) is called open- circuit.  12

12. E xa m p le The power absorbed by the 10-kΩ resistor in the following circuit is 3.6 mW. Determine the voltage and the current in the circuit. 13

13. E xa m p le Given the following network, find R and V S . 14

14. E xa m p le Given the following circuit, find the value of the voltage source and the power absorbed by the resistance. 15

15. Terminology (Nodes and Branches) Please note that almost all components that we deal with in this course are two-terminal components (resistors, sources, …) node ” (or node for short) is the point of connection of 16 A t h r ee o r m ore c i r c uit ele m en t s . ( T he n ode i n c lud e s t he interconnection wires.) A “ binary node ” (or b-node for short) has only two components connected to it.

16. E xa m p le In the following circuit identify the nodes (and their types). 24

17. E xa m p le Are the following two circuits different? Identify the nodes (and their types) in each circuit. 18

18. E xa m p le Are the following two circuits different? Identify the nodes (and their types) in each circuit.

19. Branch A branch is a collection of elements that are connected between two “ nodes ” that includes only those two true nodes (and does not include any other true nodes). In our example: 20

20. Loo p A “ loop ” is any closed path in the circuit that does not cross any true node but once. A “ window pane loop ” is a loop that does not contain any other loops inside it. A n “ i nde p e n dent l oop” is a lo o p t hat c on t ai n s at lea s t one branch that is not part of any other independent loop. 21

21. E xa m p le In the following circuit, find the number of branches, nodes, and window pane loops. Are the window pane loops independent? 22

22. Series and Parallel Connections Two or more elements are connected “ in series ” when they belong to the same branch.(even if they are separated by other elements). In general, circuit elements are in series when they are sequentially connected end-to-end and only share binary nodes among them. Elements that are in series carry the same current. 23

23. Series and Parallel Circuits T w o or m or e c i r c uit el e m en t s a re “ i n para lle l ” i f t hey a re connected between the same two “true nodes”. Consequently, parallel elements have the same voltage 24

24. Kirchhoff’s Current Law (KCL) Gustav Robert Kirchhoff (1824-1887), a German physicist, The algebraic sum of the currents entering a node (or a closed boundary ) is zero . The algebraic sum of the currents leaving a node (or a closed boundary ) is zero . the sum of the currents entering a node is equal to the sum of the currents leaving the node. The current entering a node may be regarded as positive while the currents leaving the node may be taken as negative or vice versa. KCL is based on the law of conservation of charge.

25. The algebraic sum of the currents entering a node (or a closed boundary ) is zero. The algebraic sum of the currents leavinging a node (or a closed boundary ) is zero. the sum of the currents entering a node is equal to the sum of the currents leaving the node. KCL @ Node 3

26. KCL Example: Write the KCL for the node A inside this black box circuit: i 4 i 3 i 2 i 1 Black box circuit A

27. KCL Alternative statement of KCL: For lumped circuits, the algebraic sum of the currents leaving a node (or a closed boundary) is zero. • Can you think of another statement for KCL? The sum of the currents entering a node is equal to the sum of the currents leaving that node. Σi in =Σi out

28. E xa m p le The following network is represented by its topological diagram. Find the unknown currents in the network.

29. E xa m p le In the following circuit, find i x .

30. Closed Boundary A closed boundary is a closed curve (or surface), such as a circle in a plane (or a sphere in three dimensional space) that has a well-defined inside and outside. This closed boundary is sometimes called supernode or more formally a Gauss surface. Johann Carl Friedrich Gauss (1777-1855) German mathematician

31. KCL Example Draw an appropriate closed boundary to find I graphical circuit representation. in the following 3A 2A I

32. E xa m p le In the following circuit, use a closed surface to find I 4 .

33. Kirchhoff’s Voltage Law (KVL) KVL : The algebraic sum of the voltage drops around any closed path (or loop) is zero at any instance of time. KVL : The algebraic sum of the voltage rise around any closed path (or loop) is zero at any instance of time. Sum of voltage drops=Sum of voltage rises As we move around a loop, we encounter the plus sign first for a decrease in energy level and a negative sign first for an increase in energy level.

34. Kirchhoff’s Voltage Law (KVL) KVL : The algebraic sum of the voltage drops around any closed path (or loop) is zero at any instance of time. Write KVL for the above circuit. Sum of voltage drops=Sum of voltage rises

35. KVL Example Find V AC and V CH in the following circuit. B C G H -2V A D F + - - - - 2V + E + + 1V 4V

36. E xa m p le In the following circuit, find v o and i .

37. E xa m p le In the following circuit, assume V R1 =26V and V R2 =14V. Find V R3 .

38. E xa m p le In the following circuit use KVL to determine V ae and V ec . Note that we use the convention V ae to indicate the voltage of point a with respect to point e or V ae =V a -V e

39. Some Interesting Implications of KCL and KVL A s e r i es c onn e c t i on of t w o d i ff e r e nt c u r rent s o u r c e s i s impossible. Why? A para ll el c onn e c t i on of t w o d i ff e r e nt v o l t age s o u r c e s i s impossible. Why?

40. More Interesting Implications A current source supplying zero current is equivalent to an open circuit: A voltage source supplying 0V is equivalent to a short circuit:

41. Series Resistors The equivalent resistance of any number of resistors connected in series is the sum of the resistors (Why?) G n G eq G 1 G 2 1 1  1 1   ......  R eq  R 1  R 2  ...  R n or

42. Voltage Division Rule In a series combination of n resistors, the voltage drop across the resistor R j for j= 1,2, …, n is: What is the formula for two series resistors?! v ( t ) R 1  R 2  …  R n R j in j v ( t ) 

43. Parallel Resistors The equivalent conductance of resistors connected in parallel is the sum of their individual conductances: G eq  G 1  G 2  ….  G n or Why? n R eq R 1 R 2 R  ….  1 1  1  1

44. Current Division In a parallel combination of n resistors, the current through the resistor R j for j= 1,2, …, n is: Wh y ? i ( t ) G 1  G 2  …  G n G j i n j i ( t ) 

45. Parallel Resistors and Current Division Example For the special case of two parallel resistors Wh y ? i ( t ) R 1  R 2 R 1 2 i ( t ), and i ( t )  R 1  R 2 R 2 1 , i ( t )  R 1  R 2 R 1 R 2 eq R 

46. E xa m p le In the following circuit find R eq :

47. E xa m p le In the following circuit find the resistance seen between the two terminal s A and B, i.e., R AB •

48. E xa m p le In the following circuit find the current i.

49. E xa m p le In the following circuit find I 1 , I 2 , I 3 , V a , and V b .

50. Tricky Example! circuit, find the equivalent resistance R eq . In the following Assume g m =0.5S. g m v 1 + v 1 -

51. Standard Resistor Values for 5% and 10% Tolerances

52. E xa m p le Given the network shown in Fig. 2.31: (a) find the required value for the resistor R; (b) use Table 2.1 to select a standard 10% tolerance resistor for R; (c) using the resistor selected in (b), determine the voltage across the 3.9-kΩ resistor; (d) calculate the percent error in the voltage V1, if the standard resistor selected in (b) is used; and (e) determine the power rating for this standard component.

53. Board Notes

54. Wye-Delta Transformations In some circuits the resistors are neither in series nor in parallel. For example consider the following bridge circuit: how can we combine the resistors R 1 through R 6 ?

55. Wye and Delta Networks A u s e f ul t e c hn i que t h at c an b e u s ed t o s im p l y m any s u c h circuits is transformation from wye (Y) to delta (  ) network. A wye (Y) or tee (T) network is a three-terminal network with the following general form:

56. Wye and Delta Networks The delta (  ) or pi (  ) network has the following general form:

57. Delta-Wye Conversion In some cases it is more convenient to work with a Y network in place of a  network. Let’s superimpose a wye network on the existing delta network and try to find the equivalent resistances in the wye network

58. Delta-Wye Conversion We calculate the equivalent resistance between terminals a and c while terminal b is open in both cases: R ac ( Y )  R 1  R 3 R ac (  )  R b ( R a  R c ) Similarly: R a  R b  R c R ac ( Y )  R ac (  )  R 1  R 3  R b ( R a  R c ) R a  R b  R c R 2  R 3  R a ( R b  R c ) R a  R b  R c R 1  R 2  R c ( R a  R b )

59. Delta-Wye Conversion Solving for R 1 , R 2 , and R 3 we have: R b R c R a  R b  R c R c R a R 2  R 1  Each resistor in the Y network is the product if the resistors in the two adjacent  branches, divided by the sum of the three  resistors. R a R b R a  R b  R c R a  R b  R c R 3 

60. Wye-Delta Conversion From the previous page equations, we have: R 1 R 2  R 2 R 3  R 3 R 1  R a R b R c ( R a  R b  R c ) R a  R b  R c ( R a  R b  R c ) 2 R a R b R c  Dividing this equation by each of the previous slide equations: R a  R 1 R 2  R 2 R 3  R 3 R 1 , R b  R 1 R 2  R 2 R 3  R 3 R 1 , and R c  R 1 R 2  R 2 R 3  R 3 R 1 R 1 R 2 R 3 Each resistor in the  network is the sum of all the possible products of Y resistors taken two at a time, divided by the opposite Y resistor

61. Wye-Delta Transformations Y and  networks are said to be balanced when: R 1  R 2  R 3  R Y and R a  R b  R c  R  F or ba l a n c e d Y and  ne t w o r k s t he c o n v e r s i o n f o r m u l a s become: and R   3 R Y R Y  R  3

62. E xa m p le For the following bridge network find R ab and i .

63. E xa m p le Find I S ? R b R c R a  R b  R c R c R a R 2  R 1  R a R b R a  R b  R c R a  R b  R c R 3 

64. E xa m p le Find I S ?