Transcript CSCI 2980: Introduction to Circuits, CAD, and Instrumentation

EENG 2610: Circuit Analysis
Class 4: Nodal Analysis
Oluwayomi Adamo
Department of Electrical Engineering
College of Engineering, University of North Texas
Nodal Analysis
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Nodal analysis is a systematic method to calculate all currents and
voltages in circuits that contain multiple nodes and loops.
In nodal analysis the variables in the circuit are selected to be the
node voltages. All other unknown variables are expressed in terms
of node voltages.
One node is selected as reference node and all other node voltages
are defined with respect to the reference node
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This node is called ground, symbol:
 vR 
V2
V4
V6
Nodal Analysis
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In nodal analysis, we employ KCL equations in such a way that
the variables contained in these equations are unknown node
voltages of the network.
One of the nodes in an N-node circuit is selected as the
reference node, and node voltage at the remaining N-1 nonreference nodes are defined with respect to this reference node.
Exactly N-1 linearly independent KCL equations are needed to
determine the N-1 unknown node voltages, which means
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Once reference node is selected, our task is to identify remaining
N-1 nodes and write one KCL equation for each of the nodes.
Circuits Containing only
Independent Current Sources
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Apply KCL and Ohm’s Law
Write N-1 linearly independent KCL equations
Three techniques to solve simultaneous equations
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Gaussian elimination
Matrix analysis
Matlab software or calculator
Example 3.1: Determine node voltages and branch currents. i A  1 m A, iB  4m A
R1  12 k, R2  R3  6 k
Circuits Containing Dependent Current Sources
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Write KCL nodal equations using dependent sources as regular
sources.
For each dependent source we add one equation expressing the
controlling variable in terms of the node voltages
Example 3.3: Find io using nodal analysis.
Circuits Containing Independent Voltage Sources
Example 3.5: Determine node voltages
Example 3.6: Find currents in resistors
Super-node
We do not apply KCL at any node – even the
reference node – that contains an
independent voltage source.
Circuits Containing Dependent Voltage Sources
Example 3.10: Determine voltage VO
V1
V2
V3
V4
Problem Solving Strategy for
Nodal Analysis
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Step 1: Define node voltages
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First, select one node in the N-node circuit as the
reference node, or ground node.
Then, define (N - 1) node voltages with respect to the
reference node.
Step 2: If current sources are present
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If only independent current sources are present:
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If dependent current sources are present:
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Write KCL equations for (N - 1) non-reference nodes.
First, write KCL equations for (N – 1) non-reference nodes.
Then, write control equations for dependent current sources.
Step 3:
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<see next slide>
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Step 3: If voltage sources are present
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If voltage sources are connected between reference
node and a non-reference node
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If the source is an independent voltage source, the voltage at
the non-reference node is known, determined by the source.
If the source is a dependent voltage source, treat it as
independent source when writing KCL equation, but add a
control equation for the dependent source.
If voltage sources are connected between two nonreference nodes, we don’t write KCL for these two
nodes, instead,
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If the source is independent voltage source,
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Since the voltage between two nodes is constrained by the voltage
source, write a constraint equation to describe this relation.
The surface of the network described by the constraint equation is
called super-node. Write a KCL equation for this super-node.
If the source is dependent voltage source,
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The source is treated as independent voltage source by writing a
constraint equation and a KCL equation for the super-node.
In addition, add a control equation for the dependent source.