CH 3 Nodal and Loop Analysis Techniques PDF

Preview

Summary

Download CH 3 Nodal and Loop Analysis Techniques PDF

Description

CH 3 NODAL AND LOOP ANALYSIS TECNIQUES

In CH2, we analyzed the simplest possible circuits containing only a single-node pair or a single loop. We found that these circuits can be completely analyzed via a single algebraic equation. In the case of a single-node pair circuit, once the node voltage is known, we can calculate all the currents. In the case of a single loop circuit, once the loop current is known, we can calculate all the voltages. In this chapter, we extend our capability In a systematic manner so that we can calculate all currents and voltages in circuits that contain multiple nodes and loops. Our analyses are based on primarily on two laws with which we are already familiar: KCL and KVL. In a nodal analysis, we employ KCL to determine the node voltages and in a loop analysis we use KVL to determine the loop currents. 3.1 Nodal Analysis In the nodal analysis, the variables in the circuit are selected to be node voltages. The node voltages are defined with respect to a common point in the circuit. One node is selected as the reference node and all the other node voltages are defined with respect to that node. We can arbitrarily select any node as a reference node but it is convenient to select a node with most wires or the largest number of branches as a reference node. Nevertheless, we usually select the node at the bottom of the circuit as a reference node. The reference node is commonly called ground because it is said to be at ground-zero potential, and it sometimes represents the chassis or ground line in a practical circuit. We will select all node voltages as being positive with respect to the reference node. If one or more of the node voltages are actually negative with respect to the reference node, the analysis will indicate it.

1

NODAL ANALYSIS FOR CIRCUITS CONTAINING ONLY INDEPENDENT CURRENT SOURCES

Goal: To find Node Voltages Procedure to Analyze circuits Using NODAL Analysis for Circuits Containing only Independent Current Sources 1. Identify nodes and Number each node. 2. Select the node at the bottom of the circuit as a REFERENCE Node. 3. Assign the Node Voltages to all the nodes except the reference node. Let nn : #of nodes. Then # of Unknowns Node Voltages=

nn  1

# of Nodal Equations= n n  1 4. Write the node equation at each node other than the reference node by applying KCL to each node and expressing each current in terms of node voltage(s). i. Expressing current in terms of node voltage when a Resistor is connected between a node and reference node

ii. Expressing current in terms of node voltages when a Resistor is connected between two nodes.

2

5. Solve

nn  1 nodal equations simultaneously to find nn  1 node voltages.

Example: Write all node equations for the circuit shown below.

3

4

5

Matrix Form of Node Equations   Re ciprocal of Re sistan ces Connected to Node1   Re ciprocal of Re sistan ces Connected between Node 2 and 1  .   

 Re ciprocal of Re sistan ces Connected between Node 1 and 2  Re ciprocal of Re sis tan ces Connected to Node 2



. 

 



 Node Voltage at Node 1  =  Node Voltage at Node 2     Currents from Current Sources at Node 1     Currents from Current Sources at Node 2      

or   Conductan ces Connected to Node1  Conduc tan ces Connected between Node 2 and 1    .   

 Conductan ces Connected between Node 1 and 2  Conduc tan ces Connected to Node 2 . 

   

     

 Node Voltage at Node 1 =  Node Voltage at Node 2       Currents from Current Sources at Node 1     Currents from Current Sources at Node 2      

Here, Signs of currents of current sources :

 for currents entering node : - for currents leaving node

or

GV    I  where

G : Conductance Matrix V  : Node Voltage Vector  I  : Current Vector

6

     

Example 3.1.Suppose that the circuit above has the following parameters: 𝑰𝑨 = 𝟏𝒎𝑨, 𝑹𝟏 = 𝟏𝟐𝑲𝜴, 𝑹𝟐 = 𝟔𝑲𝜴, 𝑹𝟑 = 𝟔𝑲𝜴 and 𝑰𝑩 = 𝟒𝒎𝑨,. Determine all node voltages and branch currents.

7

8

Example: Write all node equations for the circuit shown below.

9

10

Example 3.1.Suppose that the circuit below has the following parameters: 𝑰𝑨 = 𝟒𝒎𝑨, 𝑹𝟏 = 𝟐𝑲𝜴, 𝑹𝟐 = 𝟐𝑲𝜴, 𝑹𝟑 = 𝟒𝑲𝜴, 𝑹𝟒 = 𝟒𝑲𝜴 and 𝑰𝑩 = 𝟐𝒎𝑨,. Determine all node voltages using by inspection.

11

12

13

14

15

16

17

18

NODAL ANALYSIS FOR CIRCUITS CONTAINING INDEPENDENT VOLTAGE SOURCES The approach that has been outlined above is perfectly general except for those networks that contain independent voltage sources. At first glance, it may seem that the presence of a voltage source complicates the nodal analysis. We can no longer write the node equations at the node with a voltage source because we do not know the current through the voltage sources. However, the nodal analysis is no more complicated and in many cases is even easier to apply when the voltage source is present. There are two distinct cases. 1. Independent voltage source is connected between a node and the reference node. 2. Independent voltage source is connected between two nodes other than the reference node.

19

1. Independent voltage source is connected between a node and the reference node. For the node with a voltage source, that node voltage is already known. Therefore, we can treat that node voltage as known one rather than unknown one. Let  nn : #of Nodes 

nv : #of Voltage Sources

In this case, the number of the node equations and the number of the unknown node voltages is 𝒏𝒏 − 𝒏𝒗 − 𝟏 and write 𝒏𝒏 − 𝒏𝒗 − 𝟏 node equations at the nodes where no voltage source is connected. For the node with a voltage source, the node voltage is determined from

Example: Find 𝑽𝟏 and 𝑽𝟐 using the nodal analysis. Firstly, you have to know that the current through the voltage source is not known. Therefore, you cannot write the node equation for the node 1. Secondly, the node voltage 𝑽𝟏 = 𝟏𝟎𝑽 at the node 1 is already known. Thus, there is no need to write the node equation for the node 1. In this case, 𝒏𝒏 = 𝟑 and 𝒏𝒗 = 𝟏 and the number of unknown voltages is 𝒏𝒏 − 𝒏𝒗 − 𝟏 = 𝟑 − 𝟏 − 𝟏 = 𝟏 . Thus, we will write the node equation only for the node 2.

20

Example 3.5: Find 𝑽𝟏, 𝑽𝟐 and 𝑽𝟑 using the nodal analysis.

21

22

23

24

2. Independent voltage source is connected between two nodes other than the reference node. If a voltage source is connected between two nodes as shown below, that node is said to be a supernode or generalized node. As it was stated above, the current through the voltage source is not known. Therefore, we cannot write the node equation for

the supernode. However, the node equation is determined from 𝑺𝒖𝒎 𝒐𝒇 𝑪𝒖𝒓𝒓𝒆𝒏𝒕𝒔 𝒂𝒕 𝒐𝒏𝒆 𝒆𝒏𝒅 𝒐𝒇 𝒔𝒖𝒑𝒆𝒓𝒏𝒐𝒅𝒆 = 𝑺𝒖𝒎 𝒐𝒇 𝑪𝒖𝒓𝒓𝒆𝒏𝒕𝒔 𝒂𝒕 𝒕𝒉𝒆 𝒐𝒕𝒉𝒆𝒓 𝒆𝒏𝒅 𝒐𝒇 𝒔𝒖𝒑𝒆𝒓𝒏𝒐𝒅𝒆 as it is shown below.

25

In this case, we have to also write a KVL equation around the supernode. This equation is called Supernode Voltage Constraint Equation. It is determined from

Let  

nn : #of Nodes nv : #of Voltage Sources

In this case, the number of the node equations is 𝒏𝒏 − 𝒏𝒗 − 𝟏 and the number of the unknown node voltages is equal to 𝒏𝒏 − 𝟏.

26

For each supernode, we have to write two equations, namely the supernode KCL and the supernode voltage constraint equations. Example 3.6: Find 𝑽𝟏 and 𝑽𝟐using the nodal analysis.

27

28

Now, we will see the case of the combinations of two cases. In other words, voltage sources are connected between two nodes, and a node and a reference node.

Procedure to Analyze circuits Using NODAL Analysis for Circuits Containing Voltage Sources 1. 2. 3. 4.

Identify nodes and Number each node. Select the node at the bottom of the circuit as a REFERENCE Node. Assign the Node Voltages to all the nodes except the reference node. Let  𝒏𝒏 : #𝐨𝐟 𝐍𝐨𝐝𝐞𝐬  𝒏𝒗 : #𝐨𝐟 𝐕𝐨𝐥𝐭𝐚𝐠𝐞 𝐒𝐨𝐮𝐫𝐜𝐞𝐬 Note that if a voltage source is connected between a node and a reference node, then the node voltage is already known as it is shown below.

29

If a voltage source is connected between two nodes, then it is called the Supernode as it is shown below.

The supernode voltage is determined from

5. Write 𝒏𝒏 − 𝒏𝒗 − 𝟏 node and/or supernode equations. 6. Write each supernode equation as it is shown below.

30

Express currents through resistors in terms of node voltage(s). 7. Write 𝒏𝒗 voltage constraint equations. 8. Solve all equations simultaneously to find all unknown node voltages. Example 3.7: Find 𝑽𝟏, 𝑽𝟐and 𝑰𝟎 using the nodal analysis.

31

32

33

34

35

36

37

38

39

3.2 Loop Analysis (Mesh Analysis) In the nodal analysis of the previous sections, we applied KCL at the nonreference nodes of the circuit. We shall now consider another method known as mesh analysis or loop analysis in which KVL is applied around certain closed paths in the circuit. In this case, the unknown are generally currents.

Goal: To find Mesh Currents Mesh is a special case of a loop.  Mesh is a closed path that does not encircle any element. or  Mesh is a closed path that does not contain other loops. The following circuit has three meshes as shown below.

Note that Loop1, Loop2 and Loop3 in the following circuit are not meshes, but they are loops because  Loop1 encloses 𝑹𝟐 or contains both Mesh1 and Mesh2.  Loop2 encloses 𝑹𝟐 and 𝑹𝟑 or contains Mesh1, Mesh2 and Mesh3.  Loop3 encloses 𝑹𝟑 or contains both Mesh2 and Mesh3.

40

MESH ANALYSIS FOR CIRCUITS CONTAINING ONLY INDEPENDENT VOLTAGE SOURCES Procedure to Analyze circuits Using MESH Analysis for Circuits Containing Independent Voltage Sources 6. Identify meshes and Number each mesh. 7. Assign the Mesh currents to all the meshes. 8. Let

n m : #of Meshes.

9. # of Unknowns Mesh Currents=

nm

n

10. # of Mesh Equations= m 11. Write the mesh equation around each mesh by applying KVL to each mesh and expressing voltage across each resistor in terms of mesh current(s).  Expressing voltage across a resistor in terms of mesh current(s).

12. Solve

nm

mesh equations simultaneously to find

n m mesh currents. 41

Example: Write all mesh equations for the circuit shown below.

42

Matrix Form of Mesh Equations   Resis tances around mesh 1  Resis tan ces between mesh 1 and 2   Resis tances between mesh 2 and 1   Resis tan ces around mesh 2   . .        Mesh Current around mesh 1  =  Mesh Current around mesh 2       Voltages of Voltage Sources around mesh 1   Voltages of Voltage Sources around mesh 2      

     

Here, Signs of voltages of voltage sources :  going from – to + : - going from + to or

 RI   V  where

 R : Resistance Matrix V  : Voltage Vector  I  : Mesh Current Vector

43

Example 3.12. Using the mesh analysis, find the currents 𝑰𝟏 , 𝑰𝟐, 𝑰𝒐 and the voltages 𝑽𝒐 , 𝑽𝟔𝑲 , 𝑽𝟑𝑲 in the circuit below.

44

Example 3.13.For the circuit below, determine all mesh currents using by inspection

45

E 3.13. Using the mesh analysis, determine the currents 𝑰𝟏 and 𝑰𝟐 , and the voltage 𝑽𝒐 in the circuit below.

46

47

48

49

50

51

52

MESH ANALYSIS FOR CIRCUITS CONTAINING INDEPENDENT CURRENT SOURCES The approach that has been outlined above is perfectly general except for those networks that contain independent current sources. At first glance, it may seem that the presence of a current source complicates the mesh analysis. We can no longer write the mesh equation around the mesh containing a current source because we do not know the voltage across the current sources. However, the mesh analysis is no more complicated and in many cases is even easier to apply when the current source is present. There are two distinct cases. 3. Independent current source is on the perimeter of a circuit. 4. Two adjacent meshes have an independent current source in common. 3. Independent voltage source is on the perimeter of a circuit. If a current source is on the perimeter of a circuit, then the mesh current is already known as it is shown below. In this case, the mesh current is determined from

Procedure to Analyze circuits Using MESH Analysis for Circuits Containing

Independent Current Source on Perimeter of Circuit 1. Identify meshes and Number each mesh. 2. Assign the Mesh currents to all the meshes. 3. Let 

n m : #of Meshes n

 I : #of Current Sources on perimeter of circuit 4. Write mesh equations around only meshes that do not contain current source(s). 5. Find mesh current(s) in mesh(es) with current source(s). 6. Solve all equations simultaneously to find nm  n I mesh currents.

53

Example 3.14: Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐 and the voltage 𝑽𝒐 in the circuit below. Firstly, you have to know that the voltage across the current source is not known. Therefore, you cannot write the mesh equation around the first mesh. Secondly, the mesh current 𝑰𝟏 = 𝟐𝒎𝑨 around the second mesh is already known. Thus, there is no need to write the mesh equation for the mesh 1. In this case, 𝒏𝒎 = 𝟐 and 𝒏𝑰 = 𝟏 and the number of unknown mesh current is 𝒏𝒎 − 𝒏𝑰 = 𝟐 − 𝟏 = 𝟏 . Thus, we will write the mesh equation only around the mesh 2.

54

Example 3.15: Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐 and the voltage 𝑽𝒐 in the circuit below.

55

56

57

58

59

60

61

Example : Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐 and the voltage 𝑽𝒐 in the circuit below.

62

Example : Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐 and the voltage 𝑽𝒐 in the circuit below.

63

2. Two adjacent meshes have an independent current source in common. (Independent current source is between (in common) two adjacent meshes), If two adjacent meshes have an independent current source in common as shown below, then the mesh equations for both mesh1 and 2 cannot be written because of the fact that

the voltage across the current source is not known. Instead, we write a loop equation around a loop containing the current source as shown below. The loop containing the current source is called “SUPERMESH”. Supermesh is a closed path that encircles current source(s) as shown below. or Supermesh is a larger mesh created from two or more meshes as shown below but it does not pass through any current source.

64

For each supermesh, write one equation for supermesh and at least another equation for supermesh current constraint. The supermesh current constraint equation is determined from

However, there may be a case that supermesh does not exist as shown below. In this case, write only the supermesh current constraint equation. For the following circuit, the supermesh current constraint equation 𝑰𝟏 − 𝑰𝟑 = 𝟒𝒎𝑨 Note that the mesh currents 𝑰𝟑 = −𝟐𝒎𝑨,thus, 𝑰𝟏 = 𝟐𝒎𝑨.

65

Procedure to Analyze circuits Using MESH Analysis for Circuits Containing

Current Sources shared by two adjacent meshes 1. Identify meshes and Number each mesh. 2. Assign the Mesh currents to all the meshes. 3. Let 

nm : #of Meshes



n I : #of Current Sources

4. Write nm  n I mesh equations around meshes that do not contain any current source. 5. For each supermesh, write one equation for supermesh and at least another equation for supermesh current constraint as shown below.

Supermesh Current Constraint Equations:

6. Solve all equations simultaneously to find all unknown mesh currents.

66

Example : Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐 and the voltage 𝑽𝒐 in the circuit below.

67

If there is more than one choice for a supermesh, select the one with less elements or the one with voltage sources, but you have to use only one supermesh not all of them.

68

69

70

71

Now, we will see the case of the combinations of two previous ones. In this case, a circuit may contain  At least one independent current source which is on the perimeter of a circuit and  At least two adjacent meshes which have an independent current source in common.

72

Example 3.16 : Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐, 𝑰𝟑 and the current 𝑰𝒐 in the circuit below.

73

74

75

76

77

78

79

80

MESH ANALYSIS FOR CIRCUITS CONTAINING ONLY DEPENDENT SOURCES Example 3.17. Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐 and the voltage 𝑽𝒐 in the circuit below.

81

Example 3.18. Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐, 𝑰𝟑 and the voltage 𝑽𝒐 in the circuit below.

82

Example 3.20. Using the mesh analysis, find the mesh currents 𝑰𝟏 , 𝑰𝟐, 𝑰𝟑 and 𝑰𝟒.

83

Comparison of Node and Mesh Methods The analysis of a complex circuit can usually be accomplished by either the node method or mesh method depending on the particular circuit under consideration. When one specifically needs to know a node voltage somewhere in the circuit, the node method is preferred or when one needs to know current the mesh method is favored. In some cases one method is clearly preferred over another. For example, when the circuit contains only voltage sources, it is probably easier to use the mesh method. When the circuit contains only current sources, it will usually be easier to use the node method. If a circuit has both current sources and voltage sources, it can be analyzed by either method. One approach is to compare the number of equations required for each method. If the circuit has fewer nodes than meshes, it may be wise to select the node method. If the circuit has fewer meshes than nodes, it may be easier to use the mesh method. Another point to consider when choosing between the two methods is what information is required. If you need to know several currents, it may be wise to proceed directly with mesh analysis. It is often helpful to determine which method is more appropriate for the problem requirements and to consider both methods. Example. Determine the best analysis method for the circuits shown below when it is required to determine a) The voltage 𝑽𝒂𝒃 in Figure (a). b) The current 𝑰 through resistor 𝑹𝟐 in Figure (b). c) The current 𝑰 in Figure (c).

Figure (a) The circuit of Figure (a) is most suitable for the node analysis. Since 𝑽𝟏 = 𝑽𝒔 we need only write node equation at node 2.

84

Figure (b) The circuit of Figure (b) is most suitable for the mesh analysis. Since the current source defines the current of the mesh 2, we need only write mesh equation for the mesh 1.

Figure (c) The circuit of Figure (c) has two node...