Chapter 8 – Methods of Analysis Lecture 10 by Moeen Ghiyas 05/12/2015 1

Nodal Analysis (General Approach) Super Nodes Nodal Analysis (Format 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

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. Solve resulting equations for nodal voltages

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 V 1, the voltage from node 1 to ground.

Step 3: Applying KCL -I 1 and I 2 are defined as leaving node eq (1) By Ohm’s law, where and.Putting above in KCL eq (1)

Putting above in KCL eq (1) Re-arranging we have.Substituting values

Now But from Ohm’s law we already know

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

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)

Example – Determine the nodal voltages V 1 and V 2 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

Step 3 – Apply KCL at all nodes (here only one remaining super-node) Note that the current I 3 will leave the super-node at V 1 and then enter the same super-node at V V V 2 = 2

Step 4 – Relating the defined nodal voltages to the independent voltage source (initially removed), we have V 1 – V 2 = E = 12 V(Note why not V 2 – V 1 ??) Step 5 – Solve resulting two equations for two unknowns: 0.25V V 2 = 2 V 1 – V 2 = 12

Step 5 – Solve resulting two equations for two unknowns: 0.25V V 2 = 2&V 1 – 1V 2 = 12 Here by substitution method,

Now,and The currents can be determined as

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)

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

Example – Write the nodal equations for the given network Step 1 – Choose ref node & assign voltage labels Step 2 to 4 as below

Example – Write the nodal equations for the given network Similarly for V 2,

Example – Using nodal analysis, determine the potential across the 4Ω resistor Step 1 – Choose ref node & Assign voltage labels, and redraw the network

Steps 2 to 4 as below:

Check Solution

Nodal Analysis (General Approach) Super Nodes Nodal Analysis (Format Approach) 05/12/

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