Methods of AnalysisPrepared by Engr. JP Timola, LPT

Nodal Analysis

• A general procedure for analyzing circuits using node voltages as the circuit variables.

• Application of Kirchhoff’s current law at the nonreference nodes.

• Also known as the node-voltage method.

Nodal Analysis

• Choosing node voltages instead of element voltages as circuit variables is convenient and reduces the number of equations one must solve simultaneously.

• To simplify matters, we shall assume in this section that circuits do not contain voltage sources.

• In nodal analysis, we are interested in finding the node voltages.

Steps to Determine Node Voltages

1.

• Select a node as the reference node.

• Assign voltages v1,v2,...,vn−1 to the remaining n−1 nodes.

• The voltages are referenced with respect to the reference node.

Reference Nodes

• or datum node

• assumed to have zero potential

• indicated by any of the three ground symbols

• chassis ground - used in devices where the case, enclosure, or chassis acts as a reference point for all circuits.

• earth ground - the potential of the earth is used as reference

Steps to Determine Node Voltages

2.

• Apply KCL to each of the n−1 nonreference nodes.

• Node 1,

I1 = I2 +i1 +i2• Node 2,

I2 +i2 = i3• Use Ohm’s law to express the branch

currents in terms of node voltages.

Steps to Determine Node Voltages

2.

• Current flows from a higher potential to a lower potential in a resistor.

Steps to Determine Node Voltages

2.

• Node 1,

𝐼1 = 𝐼2 +𝑣1−0

𝑅1+

𝑣1−𝑣2

𝑅2

• Node 2,

𝐼2 +𝑣1−𝑣2

𝑅2=

𝑣2−0

𝑅3

Steps to Determine Node Voltages

3. Solve the resulting simultaneous equations to obtain the unknown node voltages.

𝐼1 = 𝐼2 +𝑣1−0

𝑅1+

𝑣1−𝑣2

𝑅2

𝐼2 +𝑣1−𝑣2

𝑅2=

𝑣2−0

𝑅3

EXAMPLE 1

• Calculate the node voltages in the circuit

EXAMPLE 2

• Obtain the node voltages in the circuit

EXAMPLE 3

• Determine the voltages at the nodes

ACTIVITY

• Find the voltages at the three nonreference nodes in the circuit

Answer:

v1 = 80 V,v2 =−64 V,v3 = 156 V.

Supernode

• Consists of two nonreference nodes connected by a (dependent or independent) voltage source.

• Formed by enclosing a(dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.

Supernode

Properties of Supernode

1. The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages.

2. A supernode has no voltage of its own.

3. A supernode requires the application of both KCL and KVL.

Example

Find the node voltages

Activity

Find v and I

Answer: −0.2 V, 1.4 A.