EENG 2610: Circuit Analysis Class 4: Nodal Analysis

Oluwayomi AdamoDepartment of Electrical EngineeringCollege of Engineering, University of North Texas

Nodal Analysis Nodal analysis is a systematic method to calculate all currents and

voltages in circuits that contain multiple nodes and loops. In nodal analysis the variables in the circuit are selected to be the

node voltages. All other unknown variables are expressed in terms of node voltages.

One node is selected as reference node and all other node voltages are defined with respect to the reference node This node is called ground, symbol:

Rv

2V 4V 6V

Nodal Analysis In nodal analysis, we employ KCL equations in such a way that

the variables contained in these equations are unknown node voltages of the network.

One of the nodes in an N-node circuit is selected as the reference node, and node voltage at the remaining N-1 non-reference nodes are defined with respect to this reference node.

Exactly N-1 linearly independent KCL equations are needed to determine the N-1 unknown node voltages, which means Once reference node is selected, our task is to identify remaining N-1 nodes and write one KCL equation for each of the nodes.

Circuits Containing only Independent Current Sources Apply KCL and Ohm’s Law Write N-1 linearly independent KCL equations Three techniques to solve simultaneous equations

Gaussian elimination Matrix analysis Matlab software or calculator

kRRkR

mAimAi BA

6,12

4,1

321

Example 3.1: Determine node voltages and branch currents.

Circuits Containing Dependent Current Sources Write KCL nodal equations using dependent sources as regular

sources. For each dependent source we add one equation expressing the

controlling variable in terms of the node voltages

Example 3.3: Find io using nodal analysis.

Circuits Containing Independent Voltage SourcesExample 3.5: Determine node voltages

Example 3.6: Find currents in resistors

We do not apply KCL at any node – even the reference node – that contains an independent voltage source.

Super-node

Example 3.10: Determine voltage VO

V1

V3V2 V4

Circuits Containing Dependent Voltage Sources

Problem Solving Strategy for Nodal Analysis Step 1: Define node voltages

First, select one node in the N-node circuit as the reference node, or ground node.

Then, define (N - 1) node voltages with respect to the reference node.

Step 2: If current sources are present If only independent current sources are present:

Write KCL equations for (N - 1) non-reference nodes. If dependent current sources are present:

First, write KCL equations for (N – 1) non-reference nodes. Then, write control equations for dependent current sources.

Step 3: <see next slide>

Step 3: If voltage sources are present If voltage sources are connected between reference

node and a non-reference node If the source is an independent voltage source, the voltage at

the non-reference node is known, determined by the source. If the source is a dependent voltage source, treat it as

independent source when writing KCL equation, but add a control equation for the dependent source.

If voltage sources are connected between two non-reference nodes, we don’t write KCL for these two nodes, instead, If the source is independent voltage source,

Since the voltage between two nodes is constrained by the voltage source, write a constraint equation to describe this relation.

The surface of the network described by the constraint equation is called super-node. Write a KCL equation for this super-node.

If the source is dependent voltage source, The source is treated as independent voltage source by writing a

constraint equation and a KCL equation for the super-node. In addition, add a control equation for the dependent source.