ELECTRIC CIRCUIT ANALYSIS - I

1. ELECTRIC CIRCUIT ANALYSIS - I Chapter 8 – Methods of Analysis Lecture 10 by MoeenGhiyas

2. TODAY’S LESSON CONTENTS • Nodal Analysis (General Approach) • Super Nodes • Nodal Analysis (Format Approach)

3. Nodal Analysis (General Approach) • 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

4. Nodal Analysis (General Approach) • 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

5. Nodal Analysis (General Approach) • 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.

6. Nodal Analysis (General Approach) • 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)

7. Nodal Analysis (General Approach) • Putting above in KCL eq (1) • Re-arranging we have • . Substituting values

8. Nodal Analysis (General Approach) • Now • But from Ohm’s law we already know

9. Super Node • 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

10. Super Node • 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)

11. Super Node • 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

12. Super Node • 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

13. Super Node • 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

14. Super Node • Step 5 – Solve resulting two equations for two unknowns: 0.25V1 + 0.5V2 = 2 & V1 – 1V2 = 12 • Here by substitution method,

15. Super Node • Now, and • The currents can be determined as

16. Nodal Analysis (Format Approach) • 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)

17. Nodal Analysis (Format Approach) • 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

18. Nodal Analysis (Format Approach) • Example – Write the nodal equations for the given network • Step 1 – Choose ref node & assign voltage labels • Step 2 to 4 as below

19. Nodal Analysis (Format Approach) • Example – Write the nodal equations for the given network • Similarly for V2 ,

20. Nodal Analysis (Format Approach) • Example – Using nodal analysis, determine the potential across the 4Ω resistor • Step 1 – Choose ref node & Assign voltage labels, and redraw the network

21. Nodal Analysis (Format Approach) • Steps 2 to 4 as below:

22. Nodal Analysis (Format Approach) • Check Solution

23. Summary / Conclusion • Nodal Analysis (General Approach) • Super Nodes • Nodal Analysis (Format Approach)