Circuit Techniques for Nodal and Mesh Analysis

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 VR = 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 VX 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