1. ELECTRIC CIRCUITS ECSE-2010 Spring 2003 Class 6

2. ASSIGNMENTS DUE Today (Monday): HW #2 Due Experiment #1 Report Due Activities 6-1, 6-2, 6-3, 6-4 (In Class) Activities 6-2 and 6-3 will use an ILM Tuesday/Wednesday: Will do Experiment #2 In Class (EP-2) Activities 7-1, 7-2, (In Class) Thursday: Will do Experiment #3 In Class (EP-3) Activities 8-1, 8-2, (In Class)

3. OPEN SHOP HOURS All in JEC 4104 (Studio) Wednesdays: 9-10 am – Prof. Nagy 2-4 pm – Prof. Jennings Thursdays: 9-10 am – Prof. Millard May Add More Later:

4. REVIEW Node Equations: Technique to Solve Any Linear Circuit Label Unknown Node Voltages, v 1, v 2, v 3, etc. # Unknown Nodes = # Nodes - # Voltage Sources – 1 (Reference) Write a KCL at Each Unknown Node Voltage Sum of Currents OUT of Node = 0 Relate Currents to Node Voltages (Ohm’s Law) Will Always Get the Same Number of Equations as Unknowns Solve Linear, Algebraic Equations using any Technique that Works for You

5. ADD CONTROLLED SOURCES Controlled/Dependent Sources Always Make Things Harder: Must Now Find a Constraint Equation: Must Relate Controlling Voltage or Current to Unknown Node Voltages (or unknown Mesh Currents as we will see a little later today) Must Do By Inspection; THINK! No Systematic Way of finding the Constraint Equation Will Explore with Activity 6-1

6. ACTIVITY 6-1 Define v, i using Active Convention

7. ACTIVITY 6-1 Use Node Equations to find v 1, v 2 : 4 Nodes - 1 Voltage Source - 1 (Ref) = 2 Unknown Node Voltages: v 1, v 2 ; v s is assumed to be known Constraint Equation: Need to Relate i x to v 1, v 2 i x = (v s - v 1 ) / 5k 6i x = 1.2 v s - 1.2 v 1

8. ACTIVITY 6-1 Define v, i using Active Convention

9. ACTIVITY 6-1

11. NODE EQUATIONS WITH CONTROLLED SOURCES Find the Constraint Equation: Must Relate Controlling Voltage or Current to Unknown Node Voltages Proceed with Usual Node Equations: Write KCL’s in Usual Way Will have an Extra Step of Algebra Matrix is no longer Symmetric

12. MESH EQUATIONS Another Systematic Technique for Solving ANY Linear Circuit: Will Always Work! Not Always the Easiest Technique Can Use Either Node Equations or Mesh Equations, But Cannot Mix Usually Must Choose Which to Use Will Say More About This Later

13. EXAMPLE

14. MESH EQUATIONS Mesh Equation Procedure: Label and Define ALL Mesh Currents Mesh = “Window Pane” in Circuit Mesh Current = A Current defined as flowing all the way around a Mesh Some Circuit Elements will have more than 1 Mesh Current flowing in them Mesh Currents must satisfy KCL Must Define Both Unknown Mesh Currents and Known Currents from Current Sources May Choose Any Direction for Unknown Mesh Currents

15. EXAMPLE

16. MESH EQUATIONS Mesh Equation Procedure: Label and Define ALL Mesh Currents Unknown Mesh Currents and Currents from Current Sources # of Unknown Mesh Currents = # of Meshes - # of Current Sources; Example: 2 Meshes - 0 Current Sources = 2 Unknown Mesh Currents; i 1 and i 2 Write a KVL around Each Unknown Mesh Current Sum of Voltages due to All Mesh Currents = 0 Express v’s in terms of Mesh Currents using Ohm’s Law

17. EXAMPLE

19. MESH EQUATIONS Write KVL Around Each Unknown Mesh Current: (i 1, i 2 in Example) Go Backwards Around Current Arrow Why Backwards? => Makes terms involving Unknown Mesh Currents Positive For This Example: i 1 R 2 - i 2 R 2 + i 1 R 1 - V in = 0 i 2 R 4 + i 2 R 3 + i 2 R 2 - i 1 R 2 = 0 2 Equations, 2 Unknowns => Can Solve for i 1 and i 2

20. MESH EQUATIONS Writing a KVL around Each Unknown Mesh Current will Always Provide # of Linear, Algebraic Equations = # Unknown Mesh Currents: Can Always Solve for i 1, i 2, …. For a Large Number of Unknown Mesh Currents, usually write equations in Matrix Form to solve using MAPLE, MATLAB, Cramer’s Rule, etc.

21. CIRCUIT SOLVER An Interactive Learning Module (ILM) developed by Academy for Electronic Media 1 of Many ILM’s developed at Rensselaer http://www.academy.rpi.edu/projects/ccli Click on Circuit Solver version 2 Choose “2 Mesh” First - Then “Activity 4-3” Activity 4-3 should read Activity 6-3 Activity 6-3 is same as Activity 6-4 Will do with ILM (6-3) and by Hand (6-4)

22. 2 MESH – ACTIVITY 6-2 Same Circuit as Example: Define Mesh Currents i 1 and i 2 : Drag and Drop Mesh Currents Create KVL Around i 1 : Click to Add or Delete a Term KVL Around i 1 : i 1 R 2 - i 2 R 2 + i 1 R 1 – V in = 0 KVL Around i 2 : i 2 R 4 + i 2 R 3 + i 2 R 2 – i 1 R 2 = 0 Click to Solve for all R’s = 1 ohm v out =.2 v in

23. ACTIVITY 6-2

24. KVL Around Mesh i 1 : i 1 - i 2 + i 1 – V in = 0 =>: 2i 1 - i 2 = V in KVL Around Mesh i 2 : i 2 + i 2 + i 2 – i 1 = 0 => - i 1 + 3i 2 = 0 => - 2i 1 + 6i 2 = 0 Add: 5i 2 = V in => i 2 =.2 V in Amps v out = i 2 (1 ohm) =.2 V in Volts Checks with Series/Parallel Method

25. ACTIVITY 6-3 (4-3 in ILM) Same as Activity 6-4: First Solve using Circuit Solver: Just Write Answer Down on Paper to be Turned In: Then solve by Hand (Activity 6-4): Feel Free to Comment on Whether ILM was Helpful to You:

26. ACTIVITY 6-4

27. 5 Meshes – 2 Current Sources = 3 Unknown Mesh Currents: i i, i 2, i 3 Must Define All Mesh Currents; Known and Unknown Must be Careful with these to make sure Mesh Currents are Unique This Takes Practice!

28. ACTIVITY 6-4

29. Write KVL’s around Unknown Meshes: Mesh 1; - 10 + 9 (i 1 - i 3 ) + 7 i 1 = 0 (16) i 1 + (0) i 2 + (- 9) i 3 = 10 Mesh 2; 5 ( i 2 + 4) + 2 (i 2 - i 3 ) + 10 = 0 (0) i 1 + (7) i 2 + (- 2) i 3 = - 30 Mesh 3; 8 (i 3 - 3) + 9 (i 3 - i 1 ) + 2 (i 3 - i 2 ) + 6 (i 3 + 4) = 0 (- 9) i 1 + (- 2) i 2 + (25) i 3 = 0 KVL’s Will Always Work Can now Solve for i 1, i 2, i 3

30. ACTIVITY 6-4

31. ADD CONTROLLED SOURCES Always Makes Things Harder Must Find a Constraint Equation: Relate Controlling Voltage or Current to Unknown Mesh Currents (or Unknown Node Voltages if using Node Equations) Must Do By Inspection; THINK! Algebra becomes more difficult