Node and Mesh Analysis

Transcription

1 Node and Mesh Analysis 1 Copyright ODL Jan 2005 Open University Malaysia

2 Circuit Terminology Name Definition Node Essential node Path Branch Essential Branch Loop Mesh A point where two ore more branches meet A node where three or more branches combine A trace of the adjacent circuit elements, where no element is included more than once. A path that connects two nodes, and contains a single element such as voltage source or resistor Path that connects two nodes without passing through an essential node. A closed path in a circuit A loop that does not contain any other loops. 2 Copyright ODL Jan 2005 Open University Malaysia

3 Circuit analysis methods introduced so far Voltage-current relations: Ohm s Law Kirchoff s Current Law (KCL) Kirchoff s Voltage Law (KVL) Circuit Reduction But circuit reduction is just a way of applying Ohm s Law, KCL, and KVL to simplify the analysis by reducing the number of unknowns!

4 Example Circuit Circuit reduction techniques don t apply Large number of unknowns, if we use exhaustive application of KVL, KCL, and Ohm s Law

5 Two new analysis techniques Next: Nodal Analysis Mesh Analysis Nodal analysis and mesh analysis provide rigorous ways to define a (relatively small) set of unknowns and write the circuit governing equations in terms of these unknowns

6 Nodal analysis overview Identify independent nodes The voltages at these nodes are the node voltages Use Ohm s Law to write KCL at each independent node in terms of the node voltages Solve these equations to determine the node voltages Any desired circuit parameter can be determined from the node voltages

7 Mesh analysis overview Identify mesh loops The currents around these loops are the mesh currents Use Ohm s Law to write KVL around each loop in terms of the mesh currents Solve these equations to determine the mesh currents Any desired circuit parameter can be determined from the mesh currents

8 Important observation Nodal analysis and mesh analysis are not fundamentally new analysis techniques We are still applying KVL, KCL, and Ohm s Law! Nodal and mesh analysis simply allow us to identify a reduced set of unknowns which completely characterize the circuit we can write and solve fewer equations to simplify our analysis!

9 Nodal Analysis i 1 v 1 v 2 R and i 2 v 2 v 1 R

10 Nodal Analysis We will illustrate the nodal analysis technique in the context of an example circuit:

11 Nodal Analysis Step 1: Identify a reference node Label the reference node voltage as V R = 0V The reference node is arbitrary! You are merely identifying the node to which all subsequent voltages will be referenced

12 Nodal Analysis Step 2: Kill sources and identify independent nodes Short-circuit voltage sources Open-circuit current sources The remaining nodes are independent Label voltages at these nodes

13 Nodal Analysis Step 3: Replace sources and label constrained voltages The constrained voltages are at dependent nodes Voltage sources constrain the difference in voltage between nodes they interconnect

14 Nodal Analysis Step 4: Apply KCL at each independent node

15 Nodal Analysis Step 5: Use Ohm s Law to write the KCL equations in terms of node voltages

16 Nodal Analysis Step 5: continued

17 Nodal Analysis Step 6: Solve the system of equations to determine the node voltages The node voltages can be used to determine any other desired parameter in the circuit

18 Nodal Analysis checking results Checking results in step 5: In general, in the equation for node X, the multiplicative factor on the node voltage V X will be the sum of the conductances at node X The multiplicative factors on all other node voltages in the equation will be the negative of the conductances between node X and the respective node voltage

19 Nodal Analysis checking results

20 Nodal Analysis shortcuts It is common to combine steps 4 and 5 Apply KCL and Ohm s Law simultaneously You can, if you wish, choose your current directions independently each time you apply KCL For example, you can assume that all currents are leaving the node, each time you apply KCL

21 Shortcuts applied to our example Previous Results:

22 Obtain values for the unknown voltages across the elements in the circuit below. At node 1 At node 2 v 2 v 1 v v 5 v v (-1.4)

23 (a) The circuit of Example 4.2 with a 22-V source in place of the 7-W resistor. (b) Expanded view of the region defined as a supernode; KCL requires that all currents flowing into the region must sum to zero, or we would pile up or run out of electrons. At node 1: v1 v2 v1 v At the supernode: v2 v1 v3 v1 v3 v

24 Determine the node-to-reference voltages in the circuit below.

25 Examples of planar and nonplanar networks; crossed wires without a solid dot are not in physical contact with each other.

26 (a) The set of branches identified by the heavy lines is neither a path nor a loop. (b) The set of branches here is not a path, since it can be traversed only by passing through the central node twice. (c) This path is a loop but not a mesh, since it encloses other loops. (d) This path is also a loop but not a mesh. (e, f) Each of these paths is both a loop and a mesh.

27 The Node Analysis The node analysis uses the voltages at the nodes as circuit variables. Example 3.1 Determine the voltage v 1 and v 2 using node analysis. 1 W 1 2 W V 5 W v 1 v 2 10 W 2 A Copyright ODL Jan 2005 Open University Malaysia

28 Solution The circuit shown in Figure 3.3 contains 3 essential nodes (ne = 3); so we need (ne -1) node-voltage equations to describe the circuit. The steps in node analysis are as follows: A reference node is chosen. Normally, the node with the largest number of branches will be chosen as the reference node. In Figure 3.3, the node at the bottom of the circuit (indicated by ) contained the largest number of branches, so it was chosen as the reference node. The rest of the nodes in the circuit are called non-reference nodes. Using Ohm s law, we formulate the node voltage equation for each node. For this circuit, we define the node voltages as v1 and v2. The node voltage is defined as the voltage increase from the reference node to the nonreference node. 28 Copyright ODL Jan 2005 Open University Malaysia

29 According to Kirchoff s Current Law, the total current leaving each branch is equal to zero. Therefore the node voltage equation at node 1 is: And at node 2: v1 10 v1 v1 v v 2 v1 v Solve the simultaneous equations to obtain the unknown node voltages, in this case v 1 and v 2. The simultaneous equations which describe the circuit above in terms of v 1 and v 2. By solving them, we obtain: v 1 = 9.09 V v 2 = V Copyright ODL Jan 2005 Open University Malaysia

30 Mesh Current Analysis R and V 2 I 2 I 1 V 1 I 1 I 2 R

31 Determine the two mesh currents, i 1 and i 2, in the circuit below. For the left-hand mesh, i ( i 1 - i 2 ) = 0 For the right-hand mesh, 3 ( i 2 - i 1 ) + 4 i 2-10 = 0 Solving, we find that i 1 = 6 A and i 2 = 4 A. (The current flowing downward through the 3-W resistor is therefore i 1 - i 2 = 2 A. )

32 Find the three mesh currents in the circuit below. Creating a supermesh from meshes 1 and 3: ( i 1 - i 2 ) + 3 ( i 3 - i 2 ) + 1 i 3 = 0 [1] Around mesh 2: 1 ( i 2 - i 1 ) + 2 i ( i 2 - i 3 ) = 0 [2] Finally, we relate the currents in meshes 1 and 3: i 1 - i 3 = 7 [3] Rearranging, i 1-4 i i 3 = 7 [1] -i i 2-3 i 3 = 0 [2] i 1 - i 3 = 7 [3] Solving, i 1 = 9 A, i 2 = 2.5 A, and i 3 = 2 A.

33 Mesh Analysis Figure 3.11 illustrates the example of using mesh current with the exist of dependent sources. This circuit contains six branches where the current is unknown and four nodes. Therefore three mesh current is required to describe the circuit (6 (4 1) = 3). These currents are shown in Figure Copyright ODL Jan 2005 Open University Malaysia

34 The current equations are: 50 = 5(i1 i2) + 20(i1 i3) (3.18) 0 = 5(i2 i1) + 1i2 + 4(i2 i3) (3.19) 0 = 20(i3 - i1) + 4(i3 i2) - 15iΦ (3.20) If the current that controls the independent source is expressed in terms of mesh current, i Φ = i 1 i 3 (3.21) By substituting equation (3.21) into equation (3.20) and rearranging the variables, 50 = 25i 1-5i 2-20i 3 (3.22) 0 = -5i i 2-4i 3 (3.23) 0 = -35i 1-4i 2 + 9i 3 (3.24) Solving the simultaneous equations, i 1 = A, i 2 = -1.43A and i 3 = A. 34 Copyright ODL Jan 2005 Open University Malaysia

35 Summary The number of equations is equal to the number of unknown. The Node voltage analysis is based on Kirchhoff s Current Law. The Mesh analysis is based on Kirchhoff s Voltage Law. 35 Copyright ODL Jan 2005 Open University Malaysia

36 Thank You 36 Copyright ODL Jan 2005 Open University Malaysia