Transcript Lecture 1 - Digilent Inc.

Lecture 7
•Review:
•Circuit techniques to date
•Overview of Nodal and Mesh analysis
•Nodal Analysis
•Related educational materials:
–Chapter 3.1, 3.2
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!
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