1. Nodes and reference nodes Steps of Nodal Analysis Supernodes Examples Lecture 5. Nodal Analysis 1

2. 2 Circuit Analysis – A Systematic Approach We’ve learned several tricks to perform circuit analysis: Single loop circuits, equivalent resistor, superposition etc. Nodal Analysis is a rather general method that allows you to analyze virtually all the linear circuits via a well defined recipe.

3. 3 Learning by Examples: A Summing Circuit The output voltage V of this circuit is proportional to the sum of the two input currents I 1s and I 2s. This circuit could be useful in audio applications or in instrumentation. The output of this circuit would probably be connected to an amplifier. Can you solve this problem using superposition method? + - V 500  1k  500  I 1s I 2s

4. 4 Nodal Analysis: The Recipe 1.Choose a reference node and assign 0 voltage to it. 2.Assign node voltages to the other nodes. 3.Express currents in terms of node voltages. 4.Apply KCL to each node other than the reference node. 5.Solve the resulting system of linear equations.

5. 5 Step 1. Reference Node The reference node is often called the ground node + – V 500  1k  500  I 1s I 2s

6. 6 Nodal Analysis: The Recipe 1.Choose a reference node and assign 0 voltage to it. 2.Assign node voltages to the other nodes. 3.Express currents in terms of node voltages. 4.Apply KCL to each node other than the reference node. 5.Solve the resulting system of linear equations.

7. 7 Step 2. Node Voltages V 1, V 2, and V 3 are unknowns for which we solve using KCL. 500  1k  500  I 1s I 2s 123 V1V1 V2V2 V3V3

8. 8 Nodal Analysis: The Recipe 1.Choose a reference node and assign 0 voltage to it. 2.Assign node voltages to the other nodes. 3.Express currents in terms of node voltages. 4.Apply KCL to each node other than the reference node. 5.Solve the resulting system of linear equations.

9. 9 Step 3. Currents and Node Voltages 500  V1V1 V1V1 V2V2

10. 10 Nodal Analysis: The Recipe 1.Choose a reference node and assign 0 voltage to it. 2.Assign node voltages to the other nodes. 3.Express currents in terms of node voltages. 4.Apply KCL to each node other than the reference node. 5.Solve the resulting system of linear equations.

11. 11 Step 4. KCL at Node 1 500  I 1s V1V1 V2V2

12. 12 Step 4. KCL at Node 2 500  1k  500  V2V2 V3V3 V1V1

13. 13 Step 4. KCL at Node 3 500  I 2s V2V2 V3V3

14. 14 Nodal Analysis: The Recipe 1.Choose a reference node and assign 0 voltage to it. 2.Assign node voltages to the other nodes. 3.Express currents in terms of node voltages. 4.Apply KCL to each node other than the reference node. 5.Solve the resulting system of linear equations.

15. 15 Step 5. Solving the Equations The left side of the equation is a sum of a linear combination of node voltages (variables to be determined). The right side of the equation is a sum of currents from sources entering the node. Re-organize the Equations

16. 16 Matrix Notation The three equations can be combined into a single matrix/vector equation. The equation can be written in matrix-vector form as Av = i The solution to the equation can be written as v = A -1 i

17. 17 Solving the Equation with MATLAB I 1s = 3mA, I 2s = 4mA >> A = [1/500+1/500 -1/500 0; -1/500 1/500+1/1000+1/500 -1/500; 0 -1/500 1/500+1/500]; >> i = [3e-3; 0; 4e-3]; >> v = inv(A)*i v = 1.3333 1.1667 1.5833

18. 18 + – V 500  1k  500  I 1s I 2s A General Solution Solution: V = 167I 1 + 167I 2 Can you prove this?

19. 19 Another Example: A Linear Large Signal Equivalent to a Transistor 5V 100I b + – VoVo 50  IbIb 2k  1k  +–+– + – 0.7V

20. 20 Nodal Analysis: The Recipe 1.Choose a reference node and assign 0 voltage to it. 2.Assign node voltages to the other nodes. 3.Express currents in terms of node voltages. 4.Apply KCL to each node other than the reference node. 5.Solve the resulting system of linear equations.

21. 5V 100I b + – VoVo 50  IbIb 2k  1k  0.7V 1 234 V1V1 V2V2 V3V3 V4V4 +–+– + – Another Example: A Linear Large Signal Equivalent to a Transistor

22. 22 Nodal Analysis: The Recipe 1.Choose a reference node and assign 0 voltage to it. 2.Assign node voltages to the other nodes. 3.Express currents in terms of node voltages. 4.Apply KCL to each node other than the reference node. 5.Solve the resulting system of linear equations.

23. 23 KCL @ Node 4 5V 100I b + – VoVo 50  IbIb 2k  1k  0.7V 1 234 V1V1 V2V2 V3V3 V4V4 +–+– + – Node 1: Node 4:

24. 24 How to Treat the Dependent Source We must express I b in terms of the node voltages: Equation from Node 4 becomes

25. 25 How to Deal With Nodes 2 and 3? The 0.7-V voltage supply makes it impossible to apply KCL to nodes 2 and 3, since we don’t know what current is passing through the supply. We do know that V 2 – V 3 = 0.7 V We need another equation!

26. 26 100I b + – VoVo 50  IbIb 2k  1k  0.7V 1 4 V1V1 V2V2 V3V3 V4V4 +–+– + – Supernode And don’t forget If a voltage source is not connected to the reference node, then it is supernode!

27. 27 Nodal Analysis: The Recipe 1.Choose a reference node and assign 0 voltage to it. 2.Assign node voltages to the other nodes. 3.Express currents in terms of node voltages. 4.Apply KCL to each node other than the reference node. 5.Solve the resulting system of linear equations.

28. 28 Step 5. Solving the Equations Great, one variable is already known! Write the equations for V 2, V 3 and V 4:

29. 29 Class Examples Drill Problems P2-8 and P2-10