Nodal Analysis

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

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. Nodal Analysis Steps • Select one of the n nodes as a reference node (that we define to be zero voltage, or ground). Assign voltages v1, v2, … vn-1 to the remaining n-1 nodes. These voltages are referenced with respect to the reference node. • 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. • Solve the resulting simultaneous equations to obtain the node voltages v1, v2, … vn-1.

4. Example Select a reference node as ground. Assign voltages v1, v2, and v3 to the remaining 3 nodes.

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. Example Now express i1, i2, …i5 in terms of v1, v2, v3 (the node voltages). Note that current flows from a higher to a lower potential.

7. Node 1: Node 2: Node 3:

8. Node 1: Node 2: Node 3:

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

10. 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

11. Writing the Nodal Equations by Inspection • The matrix G is symmetric, gkj = gjk and all of the off-diagonal terms are negative or zero. The gkk terms are the sum of all conductances connected to node k. The gkj terms are the negative sum of the conductances connected to BOTH node k and node j. The ik (the kth component of the vector i) = the algebraic sum of the independent currents connected to node k, with currents entering the node taken as positive.

12. MATLAB Solution of Nodal Equations

13. Test with numbers v1 v2 v3

14. MATLAB Run v1 v2 v3

15. PSpice Simulation MATLAB:

16. What happens if we have dependent current sources in the circuit? • Write the nodal equations in the same way we did for circuits with only independent sources. Temporarily, consider the dependent sources as being independent. • Express the current of each dependent source in terms of the node voltages. • Rewrite the equations with all node voltages on the left hand side of the equality.

17. Example Write nodal equations by inspection.

18. Example

19. Example

20. Example

21. Note: the G matrix will no longer necessarily be symmetric

22. 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 through the voltage source is known and write the equations by inspection.

23. Example Assume temporarily that i2 is known and write the equations by inspection.

24. There appears to be 4 unknowns (v1, v2, v3, and i2) and only 3 equations. However, from the circuit or so we can replace v1 (we could also replace v2) and write

25. Writing the above equation with the unknowns (v2, v3, i2) on the LHS yields

26. Test with numbers v1 v2 v3 Noting that

27. Test with numbers v1 v2 v3 Unknowns:

28. MATLAB Run v1 v2 v3 v2 V v3 V i2 A

29. PSpice Simulation v2 MATLAB: v3 i2