1. Lesson 8: Nodal Analysis 1

2. Learning Objectives Apply Ohm’s Law using nodal voltages.
Apply the Nodal Analysis method to determine an unknown node voltage and branch currents in a simple DC circuit.

3. Nodal Analysis
Based on a systematic application of KCL, nodal analysis allows us to obtain a set of simultaneous equations to solve for voltages and current in any linear circuit.

4. Nodal Analysis Steps
In nodal analysis we are interested in finding node voltages.
Given a circuit with n nodes:
Select a reference node.
Assign voltages va, vb, vc …. to the remaining nodes.
Apply KCL to each node expressing branch currents in terms of node voltages.
Solve the resulting simultaneous equations to obtain unknown node voltages.

5. 1. Identify the Nodes
What are the voltages of the three nodes a,b,c below?
=> Follow the steps to find….

6. 2. Write Equations for Branch Currents
In nodal analysis, we usually write the branch currents directly in terms of node voltages and branch resistances.
Care must be taken in keeping the polarity correct!
You can make the problem a little easier by always arbitrarily assuming current leaves each node.
This simplifies writing the resulting equations and prevents polarity errors.

7. 3. Substitute Into KCL and Solve for the Unknowns

8. Now Solve With Simple Algebra…

9. Example Problem 1 Solve for node voltage Vb Identify the nodes:
Draw your current arrows leaving node b.
Write known equations:
=>KCL
Solve the algebra:

10. Example Problem 2 Solve for node voltage Vb Identify the nodes:
Draw your current arrows leaving node b.
Write known equations:
=>KCL
Solve the algebra:

11. Making it Tougher…
The polarity of the voltage supplies changes how you write the equations.
Notice the inverted supply on the right and how you write the equation.

12. Again, Solve With Simple Algebra…
Solve for node voltage Vb

13. Example Problem 3
How many nodes in the circuit below have unknown voltages?
Determine IUNK.
a) 1, node b. I3 I1
b) Identify the nodes: =>
Draw your current arrows leaving node b. I2
Write known equations:
Now find Iunk:
Iunk = -I3 = 185mA
=>KCL
Solve the algebra:

14. Example Problem 4 Find the voltage at node c and the unknown current.
Identify the nodes: a, b, c and d
Draw your current arrows leaving node c. I3 I1
Write known equations: I2
KCLC = I1+I2+I3=0
Solve for the Node C Voltage:
Now find Iunk:

15. QUESTIONS?