Lecture 8•Review:

•Nodal analysis•Supernodes•Additional nodal analysis examples

•Mesh Analysis•Related educational modules:

–Sections 1.6.2

Review: Nodal Analysis

1. Choose reference node2. Identify independent nodes3. Label “constrained” voltages4. Apply KCL at independent nodes5. Write the KCL equations in terms of node voltages6. Solve equations to determine the node voltages7. Determine desired circuit parameters from node

voltages

Supernodes

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

V=0

Supernode

2mA

+ -3V

3kW 6kWi1 i2

Supernodes – continued

• A node is defined as having a single, unique voltage

• We can, however, apply KCL at supernodes which contain multiple nodes

• Example:

A Bi1 i2 i3

i4 i5

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

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:

2mA

+ -3V

3kW 6kW

Lecture 7, Example 3 – alternate approach

2mA

+ -3V

3kW 6kW

Example 1• Determine the voltage across the 6W resistor

Example 1 – alternate approach

Example 2• Use nodal analysis to write a set of equations from which

you can determine the current through the 6W resistor.

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

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

Mesh Analysis

• We will illustrate the mesh analysis technique in the context of an example circuit:

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

•

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

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

Mesh Analysis• Step 3: continued –

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

•