Chapter 8 – Methods of Analysis

Lecture 10

by Moeen Ghiyas

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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: 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

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.

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)

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 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

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

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

Step 5 – Solve resulting two equations for two unknowns:

0.25V1 + 0.5V2 = 2 & V1 – 1V2 = 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 V2 ,

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)

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