Nodal Analysis ∆ to Y and Y to ∆ Conversions Assignment # 5 13/08/2015 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: V 1, V 2, 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:
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 E 1 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 µV x with short circuit And apply KCL at node V 1 :
And apply KCL at node 2: No eqn for node 2 because V 2 is is part of reference node Revert to original circuit and make eqn Note that because the impedance Z 3 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 Z T of the network of fig Solution:Converting the upper Δ of bridge configuration to Y.
EXAMPLE - Find the total impedance Z T of the network of fig