Lecture 8 Review: Nodal Analysis, Supernodes, Mesh Analysis

1. Lecture 8 Review: Nodal analysis Supernodes Additional nodal analysis examples Mesh Analysis Related educational materials: Chapter 3.2, 3.3

2. Review: Nodal Analysis • Choose reference node • Identify independent nodes • Label “constrained” voltages • Apply KCL at independent nodes • Write the KCL equations in terms of node voltages • Solve equations to determine the node voltages • Determine desired circuit parameters from node voltages

3. Supernodes • In example 3 of lecture 7, we applied KCL at a supernode

4. Supernodes – continued • A node is defined as having a single, unique voltage • We can, however, apply KCL at supernodes which contain multiple nodes • Example:

5. Supernodes in nodal analysis • Supernodes are especially useful in nodal analysis when dependent nodes (voltage sources) are present • Define a supernode containing the dependent nodes • The supernode contains the voltage source and the nodes to which it is connected • Apply KCL at the supernode

6. Supernodes are useful, but not required • Supernodes are not essential for nodal analysis, as long as you account for all currents • Need to explicitly include currents through voltage sources • Lecture 7, Example 3:

7. Lecture 7, Example 3 – alternate approach

8. Example 1 • Determine the voltage across the 6 resistor

9. Example 1 – alternate approach

10. Example 2 • Use nodal analysis to write a set of equations from which you can determine the current through the 6 resistor.

11. Mesh analysis – review • 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

12. Nodal and mesh analysis – comparison • Nodal analysis: • Define independent nodes • Apply KCL at independent nodes • Use Ohm’s Law to write KCL in terms of node voltages • Mesh analysis: • Define “mesh loops” • Apply KVL around the mesh loops • Use Ohm’s Law to write KVL in terms of mesh currents

13. Mesh Analysis • We will illustrate the mesh analysis technique in the context of an example circuit:

14. Mesh Analysis • Step 1: Choose mesh loops and identify mesh currents • Kill sources (short voltage sources, open-circuit current sources) • Recommendation: mesh loops should not have other loops in their interior

15. Mesh Analysis • Step 2: Replace sources and write constrained loops • Constrained loops go through current sources • Constrained loops are somewhat arbitrary, but their direction and magnitude must be consistent with the source through which they pass

16. Mesh Analysis • Step 3: Apply KVL around the mesh loops • Use Ohm’s Law to write voltage drops in terms of mesh currents • Voltage polarities in KVL must be consistent with that loop’s mesh current

17. Mesh Analysis • Step 3: continued

18. Mesh Analysis • Step 4: Solve the equations for mesh currents • Use mesh currents to determine the circuit parameters of interest • Note: The total current in an element is the sum of the mesh currents in the element