chapter 8 – methods of analysis lecture 10 by moeen ghiyas 05/12/2015 1
TRANSCRIPT
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Chapter 8 – Methods of Analysis
Lecture 10
by Moeen Ghiyas
18/04/23 1
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Nodal Analysis (General Approach)
Super Nodes
Nodal Analysis (Format Approach)
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Mesh Analysis employs KVL
While Nodal Analysis uses KCL for solution
A node is defined as a junction of two or more branches
Define one node of any network as a reference (that is, a
point of zero potential or ground), the remaining nodes of the
network will all have a fixed potential relative to this
reference
For a network of N nodes, therefore, there will exist (N – 1)
nodes with a fixed potential relative to the assigned reference
node
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Steps
Determine the number of nodes within the network
Pick a reference node, and label each remaining node with a
subscripted value of voltage: V1, V2, and so on
Apply Kirchhoff’s current law at each node except the reference
Assume that all unknown currents leave the node for each
application of KCL.
Solve resulting equations for nodal voltages
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Apply nodal analysis to the network of Fig
Step 1 – The network has two nodes
Step 2 – The lower node is defined as the
reference node at ground potential (zero
volts), and the other node as V1, the
voltage from node 1 to ground.
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Step 3: Applying KCL - I1 and I2 are defined as leaving
node
------- eq (1)
By Ohm’s law, where
and
. Putting above in KCL eq (1)
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Putting above in KCL eq (1)
Re-arranging we have
. Substituting values
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Now
But from Ohm’s law we already know
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In nodal analysis technique, if voltage source is found
in the circuit, it is better to convert it to current source
and apply nodal analysis method
Concept of super node becomes applicable when
voltage sources (without series resistance) are present
in the network
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Steps
Assign a nodal voltage to each independent node, including the
voltage sources, as if they were resistors or voltage sources
Remove the voltage sources (replace with short-circuit )
Apply KCL to all the remaining independent nodes
Relate the chosen node to the independent node voltages of the
network, and solve for the nodal voltages
Any node including the effect of elements tied only to other
nodes is referred to as a super-node (since it has an additional
number of terms)
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Example – Determine the nodal voltages V1 and V2 of Fig (using the
concept of a super-node)
Step 1 - Assign Nodal Voltages
(All unknown currents leave node)
• Step 2 – Replace Voltage source
with short circuit
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Step 3 – Apply KCL at all nodes (here only one remaining super-node)
Note that the current I3 will leave the super-node at V1 and then enter
the same super-node at V2.
0.25V1 + 0.5V2 = 2
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Step 4 – Relating the defined nodal voltages to the independent
voltage source (initially removed), we have
V1 – V2 = E = 12 V (Note why not V2 – V1 ??)
Step 5 – Solve resulting two equations for two unknowns:
0.25V1 + 0.5V2 = 2
V1 – V2 = 12
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Step 5 – Solve resulting two equations for two unknowns:
0.25V1 + 0.5V2 = 2 & V1 – 1V2 = 12
Here by substitution method,
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Now, and
The currents can be determined as
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This technique allows us to write nodal eqns rapidly
A major requirement, however, is that all voltage
sources must first be converted to current sources
before the procedure is applied
Quite similar to mesh analysis (format approach)
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Choose a reference node and assign a subscripted voltage label to
(N - 1) remaining nodes of the network
Column 1 of each eqn is summing the conductances with node of
interest and multiplying the result by that node voltage
Each mutual term is the product of the mutual conductance and the
other nodal voltage and are always subtracted from the first column
The column to the right of the equality sign is the algebraic sum of
the current sources tied to the node of interest. A current source is
assigned a positive sign if it supplies current to a node and a
negative sign if it draws current from the node
Solve the resulting simultaneous equations for the desired voltages
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Example – Write the nodal equations for the given network
Step 1 – Choose ref node & assign voltage labels
Step 2 to 4 as below
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Example – Write the nodal equations for the given network
Similarly for V2 ,
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Example – Using nodal analysis, determine the potential across the
4Ω resistor
Step 1 – Choose ref node & Assign voltage labels, and redraw the
network
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Steps 2 to 4 as below:
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Check Solution
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Nodal Analysis (General Approach)
Super Nodes
Nodal Analysis (Format Approach)
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