Lecture 8
•Review:
•Nodal analysis
•Supernodes
•Additional nodal analysis examples
•Mesh Analysis
•Related educational modules:
–Sections 1.6.2
Review: Nodal Analysis
1.
2.
3.
4.
5.
6.
7.
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
Supernodes
• In example 3 of lecture 7, we applied KCL at a
supernode
Supernode
3V
+2mA
i1
3kW
i2
6kW
V=0
Supernodes – continued
• A node is defined as
having a single,
unique voltage
• We can, however,
apply KCL at
supernodes which
contain multiple
nodes
• Example:
i1
i2
i3
B
A
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:
3V
+-
2mA
3kW
6kW
Lecture 7, Example 3 – alternate approach
3V
+2mA
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:
• Mesh analysis:
• Define independent
nodes
• Define “mesh loops”
• Apply KCL at
independent nodes
• Apply KVL around the
mesh loops
• Use Ohm’s Law to write
KCL in terms of node
voltages
• 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, opencircuit 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
•