1. 1 Nodal Analysis Discussion D2.3 September 2006 Chapter 2 Section 2-7

2. 2 Nodal Analysis Interested in finding the NODE VOLTAGES, which are taken as the variables to be determined For simplicity we start with circuits containing only current sources

3. 3 Nodal Analysis Steps 1.Select one of the n nodes as a reference node (that we define to be zero voltage, or ground). Assign voltages v 1, v 2, … v n-1 to the remaining n-1 nodes. These voltages are referenced with respect to the reference node. 2.Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of the node voltages. 3.Solve the resulting simultaneous equations to obtain the node voltages v 1, v 2, … v n-1.

4. 4 Example Select a reference node as ground. Assign voltages v 1, v 2, and v 3 to the remaining 3 nodes.

5. 5 Example Apply KCL to each of the 3 non-reference nodes (sum of currents leaving node is zero). Node 1: Node 2: Node 3:

6. 6 Example Now express i 1, i 2, …i 5 in terms of v 1, v 2, v 3 (the node voltages). Note that current flows from a higher to a lower potential.

7. 7 Node 1: Node 2: Node 3:

8. 8 In MATLAB, if then is a row matrix of the five conductances

9. 9 Node 1: Node 2: Node 3:

10. 10 These three equations can be written in matrix form as

11. 11 is an (n –1) x (n –1) symmetric conductance matrix is a 1 x (n-1) vector of node voltages is a vector of currents representing “known” currents

12. 12 Writing the Nodal Equations by Inspection The matrix G is symmetric, G kj = G jk and all of the off-diagonal terms are negative or zero. The k i (the i th component of the vector k) = the algebraic sum of the independent currents connected to node i, with currents entering the node taken as positive. The G kj terms are the negative sum of the conductances connected to BOTH node k and node j. The G kk terms are the sum of all conductances connected to node k.

13. 13 MATLAB Solution of Nodal Equations

14. 14 v1v1 v2v2 v3v3 Test with numbers

15. 15 MATLAB Run v1v1 v2v2 v3v3

16. 16 PSpice Simulation MATLAB:

17. 17 Let's write a general MATLAB program to solve this problem Inputs: Find all voltages and currents

18. 18 function nodal1(r,k) % PowerPoint nodal example % Discussion D2.3 % r is a 1 x 5 vector of resistances % k is a 3 x 1 vector of known currents entering the three nodes % nodal1(r,k) % g = 1./ r G = [g(1)+g(2) -g(2) 0; -g(2) g(2)+g(3)+g(4) -g(4); 0 -g(4) g(4)+g(5)] k v = inv(G)*k i(1) = v(1)*g(1); i(2) = (v(1) - v(2))*g(2); i(3) = v(2)*g(3); i(4) = (v(2) - v(3))*g(4); i(5) = v(3)*g(5); i kab = [i(1)+i(2) i(5)-i(4)]

19. 19 Do same problem as before nodal1(r,k)

20. 20 MATLAB Run

21. 21 Nodal Analysis for Circuits Containing Voltage Sources That Can’t be Transformed to Current Sources Case 1. If a voltage source is connected between the reference node and a nonreference node, set the voltage at the nonreference node equal to the voltage of the source. Case 2. If a voltage source is connected between two nonreference nodes, assume temporarily that the current through the voltage source is known and write the equations by inspection.

22. 22 Example Assume temporarily that i 2 is known and write the equations by inspection.

23. 23 There appears to be 4 unknowns (v 1, v 2, v 3, and i 2 ) and only 3 equations. However, from the circuit or so we can replace v 1 (we could also replace v 2 ) and write

24. 24 Writing the above equation with the unknowns (v 2, v 3, i 2 ) on the LHS yields

25. 25 v1v1 v2v2 v3v3 Test with numbers Noting that

26. 26 v1v1 v2v2 v3v3 Test with numbers Unknowns:

27. 27 MATLAB Run V V A v1v1 v2v2 v3v3 v2v2 v3v3 i2i2

28. 28 PSpice Simulation MATLAB: v2v2 v3v3 i2i2

29. 29 Let's write a general MATLAB program to solve this problem Inputs: Find all voltages and currents

30. 30 function nodal2(g,V0,is) % PowerPoint nodal-2 example % Discussion D2.3 % g is a 1 x 4 vector of conductances % V0 = the known dc voltage source % is = the known dc current source % nodal2(g,V0,Is) % G = [g(1) 0 1; g(2)+g(3) -g(3) -1; -g(3) g(3)+g(4) 0] k = [-2+g(1)*V0; 0; is] vvi = inv(G)*k v = zeros(1,3); v(2) = vvi(1); v(3) = vvi(2); v(1) = v(2)-V0; v i(1) = v(1)*g(1); i(2) = vvi(3); i(3) = v(2)*g(2); i(4) = (v(2) - v(3))*g(3); i(5) = v(3)*g(4); i kab = [i(1)+i(2) i(5)-i(4)]

31. 31 Do same problem as before nodal2(g,V0,is)

32. 32 MATLAB Run