2. Objective of Lecture Provide step-by-step instructions for nodal analysis, which is a method to calculate node voltages and currents that flow through components in a circuit. Partly covered in Chapter 5.5 Electric Circuits Fundamentals Chapter 9.4 Principles of Electric Circuits Chapter 3.2 and 3.3 Fundamentals of Electric Circuits Chapter 2.4 Electrical Engineering: Principles and Applications

3. Nodal Analysis Technique to find currents at a node using Ohm’s Law, Kirchhoff’s Current Law, and the potential differences betweens nodes. First result from nodal analysis is the determination of node voltages (voltage at nodes referenced to ground). These voltages are not equal to the voltage dropped across the resistors. Second result is the calculation of the currents

4. Steps in Nodal Analysis V in

5. Steps in Nodal Analysis V in Pick one node as a reference node Its voltage will be arbitrarily defined to be zero

6. Step 1 V in Pick one node as a reference node Its voltage will be arbitrarily defined to be zero

7. Step 2 Label the voltage at the other nodes V in

8. Step 2 Label the voltage at the other nodes V in

9. Step 3 Label the currents flowing through each of the components in the circuit

10. Step 4 Use Kirchhoff’s Current Law

11. Step 5 Use Ohm’s Law to relate the voltages at each node to the currents flowing in and out of them. Current flows from a higher potential to a lower potential in a resistor The difference in node voltage is the magnitude of electromotive force that is causing a current I to flow.

12. Step 5 We do not write an equation for I 7 as it is equal to I1

13. Step 6 Solve for the node voltages In this problem we know that V1 = V in

14. Step 6 Substitute the equations obtained using Ohm’s Law into the equations obtained using KCL.

15. Step 7 Once the node voltages are known, calculate the currents.

17. From Previous Slides

18. Substituting in Numbers

19. Substituting the results from Ohm’s Law into the KCL equations

20. Chugging through the Math Node voltages must have a magnitude less than the sum of the voltage sources in the circuit One or more of the node voltages may have a negative sign This depends on which node you chose as your reference node. Node Voltages(V) V1V1 10 V2V2 5.55 V3V3 4.56 V4V4 3.74 V5V5 3.46

21. Chugging through the Math Voltage across resistors (V) V R1 = (V 1 – V 2 )4.45 V R2 = (V 2 – V 3 )0.990 V R3 = (V 3 – V 5 )1.10 V R4 = (V 3 – V 4 )0.824 V R5 = (V 4 – V 5 )0.274 V R6 = (V 5 – 0V)3.46 The magnitude of any voltage across a resistor must be less than the sum of all of the voltage sources in the circuit. In this case, no voltage across a resistor can be greater than 10V.

22. Chugging through More Math Currents (  A) I1I1 495 I2I2 I3I3 220 I4I4 275 I5I5 I6I6 495 I7I7

23. Check None of the currents should be larger than the current that flows through the equivalent resistor in series with the 10V supply. Note that this check is only valid if there is one voltage source in the circuit.

24. Summary Steps in Nodal Analysis 1. Pick one node as a reference node 2. Label the voltage at the other nodes 3. Label the currents flowing through each of the components in the circuit 4. Use Kirchhoff’s Current Law 5. Use Ohm’s Law to relate the voltages at each node to the currents flowing in and out of them. 6. Solve for the node voltage 7. Once the node voltages are known, calculate the currents.