1. Lecture 7 Review: Circuit techniques to date Overview of Nodal and Mesh analysis Nodal Analysis Related educational modules: –Sections 1.6.0, 1.6.1

2. 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!

3. Example Circuit Circuit reduction techniques don’t apply Large number of unknowns, if we use exhaustive application of KVL, KCL, and Ohm’s Law

4. 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

5. 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

6. 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

7. 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!

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

9. 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

10. 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

11. 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

12. Nodal Analysis Step 4: Apply KCL at each independent node

13. Nodal Analysis Step 5: Use Ohm’s Law to write the KCL equations in terms of node voltages –

14. Nodal Analysis Step 5: continued

15. 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

16. 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

17. Nodal Analysis – checking results

18. 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

19. Shortcuts applied to our example Previous Results:

20. Nodal analysis – example 2 Use nodal analysis to find i in the circuit below

21. Example 2 – continued

22. Example 2 – What if we mis-identify independent nodes?

23. Nodal analysis – example 3 Use nodal analysis to determine v in the circuit below

24. Example 3 – Alternate reference node