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ELECTRIC CIRCUITS ECSE-2010 Spring 2003 Class 6

ELECTRIC CIRCUITS ECSE-2010 Spring 2003 Class 6. 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)

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ELECTRIC CIRCUITS ECSE-2010 Spring 2003 Class 6

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  1. ELECTRIC CIRCUITSECSE-2010Spring 2003Class 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, v1, v2, v3, 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 v1, v2: • 4 Nodes - 1 Voltage Source - 1 (Ref) = 2 Unknown Node Voltages: • v1, v2 ; vs is assumed to be known • Constraint Equation: • Need to Relate ix to v1, v2 • ix = (vs - v1) / 5k • 6ix = 1.2 vs - 1.2 v1

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

  9. ACTIVITY 6-1

  10. 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; i1 and i2 • 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

  18. EXAMPLE

  19. MESH EQUATIONS • Write KVL Around Each Unknown Mesh Current: (i1, i2 in Example) • Go Backwards Around Current Arrow • Why Backwards? => Makes terms involving Unknown Mesh Currents Positive • For This Example: • i1 R2 - i2 R2 + i1 R1 - Vin = 0 • i2 R4 + i2 R3 + i2 R2 - i1 R2 = 0 • 2 Equations, 2 Unknowns => Can Solve for i1 and i2

  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 i1, i2, …. • 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 i1 and i2: • Drag and Drop Mesh Currents • Create KVL Around i1: • Click to Add or Delete a Term • KVL Around i1: i1 R2 - i2 R2 + i1 R1 – Vin = 0 • KVL Around i2:i2 R4 + i2 R3 + i2 R2 – i1 R2 = 0 • Click to Solve for all R’s = 1 ohm • vout = .2 vin

  23. ACTIVITY 6-2 ACTIVITY 6-2

  24. ACTIVITY 6-2 • KVL Around Mesh i1: i1 - i2 + i1 – Vin = 0 • =>: 2i1 - i2 = Vin • KVL Around Mesh i2: i2 + i2 + i2 – i1 = 0 • => - i1 + 3i2 = 0 • => - 2i1 + 6i2 = 0 • Add: 5i2 = Vin => i2 = .2 Vin Amps • vout = i2 (1 ohm) = .2 Vin 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. ACTIVITY 6-4 • 5 Meshes – 2 Current Sources = 3 Unknown Mesh Currents: • ii, i2, i3 • 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. ACTIVITY 6-4 • Write KVL’s around Unknown Meshes: • Mesh 1; - 10 + 9 (i1 - i3) + 7 i1 = 0 • (16) i1 + (0) i2 + (- 9) i3 = 10 • Mesh 2; 5 ( i2 + 4) + 2 (i2 - i3) + 10 = 0 • (0) i1 + (7) i2 + (- 2) i3 = - 30 • Mesh 3; 8 (i3 - 3) + 9 (i3 - i1) + 2 (i3 - i2) + 6 (i3+ 4) = 0 • (- 9) i1 + (- 2) i2 + (25) i3 = 0 • KVL’s Will Always Work • Can now Solve for i1, i2, i3

  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

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