Show detailed calculations and all steps leading to the final answer, include all circuit schematics used for circuit simplification used at various steps.NodeVoltageMethod.docxNodeVo
Show detailed calculations and all steps leading to the final answer, include all circuit schematics used for circuit simplification used at various steps.
28. In Figure 9–36, use the node voltage method to find the voltage at point A with respect to ground.
Figure 9–36
30. Write the node voltage equations for Figure 9–33. Use your calculator to find the node voltages.
31. Use node analysis to determine the voltage at points A and B with respect to ground in Figure 9–37.
*32. Find the voltage at points A, B, and C in Figure 9–38.
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Node Voltage Analysis
EET310 Circuit Analysis
Nodes, Branches and Loops
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Since the elements of an electric circuit can be interconnected in several ways, we need to understand some basic concepts of network topology
In network topology, we study the properties relating to the placement of elements in the network and the geometric configuration of the network.
Such elements include branches, nodes, and loops
Nodes, Branches and Loops
Branches
A branch represents a single element such as a voltage source, a current source or a resistor
In other words, a branch represents any two-terminal element
The circuit shown below has five branches, namely, the 10-V voltage source, the 2-A current source, and the three resistors.
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Nodes, Branches and Loops
Nodes
A node is the point of connection between two or more branches
A node is usually indicated by a dot in a circuit
If a short circuit connects two nodes, the two nodes constitute a single node.
The circuit shown on right has three nodes a, b, and c.
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Nodes, Branches and Loops
Loops
A loop is a closed path formed by starting at a node, passing through a set of nodes, and returning to the starting node without passing through any node more than once.
A loop is said to be independent if it contains a branch which is not in any other loop
Independent loops or paths result in independent sets of equations.
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Nodes, Branches and Loops
Loops
How many loops exist in the network shown below?
How many of them are independent loops?
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Nodes, Branches and Loops
Determine the number of branches and nodes in the network below
Determine the number of branches and nodes for the network below
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Kirchhoff’s Current Law (KCL)
Kirchhoff’s Current Law states that:
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The algebraic sum of currents entering a node (or a closed boundary) is zero.
The sum of all currents leaving a node should be the same as the sum of all currents entering the node
OR
Kirchhoff’s Voltage Law (KVL)
Kirchhoff’s Voltage Law states that:
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The algebraic sum of voltages around a closed path ( loop) is zero.
In any closed path (loop), the sum of all potential rises is equal to the sum of all potential drops
OR
Node voltage method
In the node voltage method, you can solve for the unknown voltages in a circuit using K C L.
Steps:
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Determine the number of nodes.
Select one node as a reference. Assign voltage designations to each unknown node.
Assign currents into and out of each node except the reference node.
Apply K C L at each node where currents are assigned.
Express the current equations in terms of the voltages and solve for the unknown voltages using Ohm’s law.
Node voltage method
Example:
Solve the same problem as before using the node voltage method.
Solution:
Write K C L in terms of the voltages (next slide).
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Node voltage method
Solution:
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Review & Examples Nodal Analysis
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This method uses KCL for general networks
Nodal analysis provides a general procedure for analyzing circuits using node voltages as the circuit variables
Choosing node voltages instead of element voltages as circuit variables is convenient and reduces the number of equations one must solve simultaneously
In nodal analysis, we are interested in finding the node voltages.
Nodal Analysis Procedure
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Assign
Step1: Arbitrarily assign a reference node within the circuit and indicate this node as ground
The reference node is usually located at the bottom of the circuit, although it may be located anywhere
Convert
Convert each voltage source in the network to its equivalent current source
This step, although not absolutely necessary, makes further calculations easier to understand
Assign
Arbitrarily assign voltages (V1, V2, . . . , Vn) to the remaining nodes in the circuit
these voltages will all be with respect to the chosen reference in step 1
Nodal Analysis Procedure
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Step 4: Arbitrarily assign a current direction to each branch in which there is no current source.
Using the assigned current directions, indicate the corresponding polarities of the voltage drops on all resistors
Step 5: With the exception of the reference node (ground), apply Kirchhoff’s current law at each of the nodes.
If a circuit has a total of n + 1 nodes (including the reference node), there will be n simultaneous linear equations.
Nodal Analysis Procedure
Step 6: Rewrite each of the arbitrarily assigned currents in terms of the potential difference across a known resistance.
Step 7: Solve the resulting simultaneous linear equations for the voltages (V1, V2,. . . , Vn). The simultaneous equations can be solved by using the methods we have seen in unit one
Method of substitution
Method of elimination
Matrix method using Cramer’s rule or matrix inversion in Excel
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Nodal Analysis
Example 1: Using Nodal Analysis determine the node voltages for the following circuit. Use the steps mentioned in the previous slides
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Nodal Analysis
Solution:
Step 1: Select a convenient reference node
Step 2: Convert the voltage source to equivalent current source
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Nodal Analysis
Solution Continued …
Steps 3 & 4: Arbitrarily assign node voltages and branch currents
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Nodal Analysis
Solution Continued …
Steps 5 & 6: Write the KCL equations at nodes labeled V1 and V2
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Nodal Analysis
Solution Continued …
Steps 7: Solve for the node voltages using Cramer’s rule
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Nodal Analysis
Example 2: Using Nodal Analysis determine the node voltages for the following circuit. Use the nodal analysis procedures
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Nodal Analysis
Solution:
Step 1: Select a convenient reference node
Step 2: Convert the voltage source to equivalent current source
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Nodal Analysis
Solution Continued …
Steps 3 & 4: Arbitrarily assign node voltages and branch currents
Nodal Analysis
Solution Continued …
Steps 5 & 6: Write the KCL equations at nodes labeled V1 and V2
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Nodal Analysis
Solution Continued …
Steps 7: Solve for the node voltages using Cramer’s rule
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Nodal Analysis
Example 3: Determine the node voltages using nodal analysis
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Nodal Analysis
Example 4: Using nodal analysis solve the node voltages
Nodal Analysis
Example 5: Using nodal analysis solve the node voltages
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Nodal Analysis
Example 6: Using nodal analysis solve the node voltages
Nodal Analysis
Example 7: Using nodal analysis solve the node voltages
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