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Chapter 17 – Methods of Analysis & Sel Topics
Lecture 24
by Moeen Ghiyas
19/04/23 1
Chapter 17 – Methods of Analysis & Sel Topics
Nodal Analysis
∆ to Y and Y to ∆ Conversions
Assignment # 5
19/04/23 3
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. Each node is to be treated as a separate
entity, independent of the application of KCL to the other
nodes
Solve the resulting equations for the nodal voltages
The general approach to nodal analysis includes the same
sequence of steps as for dc with minor changes to substitute
impedance for resistance and admittance for conductance in the
general procedure:
Independent Current Sources
Same as above
Dependent Current Sources
Step 3 is modified: Treat each dependent source like an
independent source when KCL is applied. However, take into
account an additional equation for the controlling quantity to
ensure that the unknowns are limited to chosen nodal voltages
Independent Voltage Sources
Treat each voltage source as a short circuit (recall the
supernode classification ), and write the nodal equations for
remaining nodes.
Relate another equation for supernode to ensure that the
unknowns of final equations are limited to the nodal voltages
Dependent Voltage Sources
The procedure is same as for independent voltage sources,
except now the dependent sources have to be defined in terms
of the chosen nodal voltages to ensure that the final equations
have only nodal voltages as the unknown quantities
EXAMPLE - Determine the voltage across the inductor for the
network of Fig
Solution:
KCL at node V1
KCL at node V2
Grouping both equations
Thus the two equations become
Solving the two equations
EXAMPLE - Write the nodal equations for the network of fig
having a dependent current source.
Solution:
Step 3 at Node 1:
EXAMPLE - Write the nodal equations for the network of fig
having a dependent current source.
At Node 1:
Step 3 at Node 2:
EXAMPLE - Write the nodal equations for the network of fig
having an independent source between two assigned nodes.
Solution:
Replacing E1 with short circuit
to get supernode circuit,
Apply KCL at node 1 or 2,
Relate supernode in nodal voltages
Solve both equations
EXAMPLE - Write the nodal
equations for the network of fig
having a dependent voltage source
between two assigned nodes.
Solution:
Replace µVx with short circuit
And apply KCL at node V1:
And apply KCL at node 2:
No eqn for node 2 because V2 is
is part of reference node
Revert to original circuit and make
eqn
Note that because the impedance Z3 is in parallel with a voltage
source, it does not appear in the analysis. It will, however, affect the
current through the dependent voltage source.
Corresponds exactly with that for dc circuits ∆ to Y,
Note that each impedance of the Y is equal to the product of the
impedances in the two closest branches of the ∆ , divided by the
sum of the impedances in the ∆.
Corresponds exactly with that for dc circuits Y to ∆,
Each impedance of the ∆ is equal to sum of the possible product
combinations of impedances of the Y, divided by the impedances
of the Y farthest from the impedance to be determined
Drawn in different forms, they are also referred to as the T
and π configurations
EXAMPLE - Find the total impedance ZT of the network of fig
Solution: Converting the upper
Δ of bridge configuration
to Y.
EXAMPLE - Find the total impedance ZT of the network of fig
EXAMPLE - Find the total impedance ZT of the network of fig
Note: Since ZA = ZB.
Therefore, Z1 =
Z2
EXAMPLE - Find the total impedance ZT of the network of fig
EXAMPLE - Find the total impedance ZT of the network of fig
. Replace the Δ by the Y
EXAMPLE - Find the total impedance ZT of the network of fig
. Solving first for series circuit
EXAMPLE - Find the total impedance ZT of the network of fig
. Resolving Parallels
. Final series solution
Ch 17 - Q 4, Q 8 (a), Q 10, Q 24
Deposit by 09:00 am Monday, 30 Apr 2012.
19/04/23 27
Nodal Analysis
∆ to Y and Y to ∆ Conversions
Assignment # 5
19/04/23 29