Ch 4: Basic Nodal and Mesh Analysis Flashcards
Nodal Analysis:
Given an N-node circuit,
How many unknown voltages?
How many equations?
A circuit with N nodes has:
- N-1 unknown voltages
- One for each node, minus the reference node
- N-1 equations
Nodal Analysis:
General Steps
-
Identify the nodes and currents
- Pick one node as the reference node(vr = 0)
- Label Node Voltages: v1, v2,…
- This is the voltage at the node with respect to the reference node
- Draw all current current arrows as pointing away from the nodes
- Label currents: i1, i2,…
-
Apply KCL
- Write the current equations for each node.
- Sum of currents leaving each node is 0
- Apply Supernode Technique if applicable
-
Apply Ohm’s Law (i = v/r)
- Rewrite the equations in terms of node voltages and resistances
-
Solve the system of equations
- Determines all node voltages
- May need to simplify first
Solving Systems of Equations:
Cramer’s Rule
- Given a system of equations:
- a1v1 + b1v2 +c1v3 = d1
- a2v1 + b2v2 +c2v3 = d2
- a3v1 + b3v2 +c3v3 = d3
- Place the coefficients in a matrix, G
- Create a matrix corresponding to each variable/column in G
- A1, A2, A3
- Each one is like G, but the column corresponding to the associated variable is replaced with the right-hand constants (d1-dn)
- Each variable can be solved by taking the ratio of determinants:
- v1 = det A1 / det G
- v2 = det A2 / det G
- v3 = det A3 / det G
Nodal Analysis:
Supernodes:
When to use? Why?
Used when there are voltage sources contained within the circuit.
A voltage source can be between the reference node and a non-reference node (The easy case)
or
A voltage source can be between two non reference nodes.
This let’s you treat the source and two nodes as a single node within the other nodal equations, while creating an additional equation relating the two nodes.
Nodal Analysis:
Supernode:
Apply to a voltage source between the reference node and a non-reference node
Simply set the voltage of the non reference node to the voltage given by the voltage source.
Nodal Analysis:
Supernode:
Apply to a voltage source between two non-reference nodes
- Circle the two nodes and voltage source, this will be the “supernode”
- Write one KCL equation for all the currents entering and exiting this “supernode”
- Write an equation for the voltage between the two nodes
- The difference is equivalent to the voltage source value
- Proceed normally with writing the equations for the rest of the nodes in the circuit
- Solve the system of equations
Mesh Analysis:
Define Mesh
A loop that does not contain any other loops within it.
Mesh Analysis:
Define Mesh Current
A current that flows only around the perimeter of a mesh
Mesh Analysis:
Define
Planar Circuit
A circuit whose diagram can be drawn on a plane surface so that no branch passes over or under another branch.
Mesh Analysis:
General Steps
- Determine if circuit is planar
- Assign mesh currents: i1, i2,…, in to each mesh
- Mesh currents should flow in the same direction, clockwise or counter clockwise
- Apply KVL
- Equations should have a term for each element
- Use Supermesh technique, if applicable
- Apply Ohm’s Law
- Rewrite the KVL equations in terms of the mesh currents and resistances
- Solve the system of equations
Mesh Analysis:
Supermesh:
When to Use? Why?
Used when two meshes share a current source as a common element.
Combines the two meshes into a “supermesh”, analytically “removing” the shared branch and creating an “internal equation” for the supermesh.
This allows the rest of the analysis to proceed as normal and provides an equation for use in solving the system of equations.
Mesh Analysis:
Supermesh:
Steps
- Draw/highlight the outline of the “supermesh”
- Should consist of the outline of the two meshes, except for the shared branch with the current source
- Write a KVL equation for this mesh(outer equation)
- Write an equation relating the current source to the mesh currents (inner equation)
- Write the rest of the equations for the other meshes in the circuit
- Solve the system of equations
