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

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!

Example Circuit

• Circuit reduction techniques don’t apply

• Large number of unknowns, if we use exhaustive application of KVL, KCL, and Ohm’s Law

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

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

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

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!

Nodal Analysis

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

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

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

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

Nodal Analysis

• Step 4: Apply KCL at each independent node

•

Nodal Analysis• Step 5: Use Ohm’s Law

to write the KCL equations in terms of node voltages

–

Nodal Analysis• Step 5: continued

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

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

Nodal Analysis – checking results

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

Shortcuts applied to our example

• • Previous Results:

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

Example 2 – continued

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

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

Example 3 – Alternate reference node