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Learning is good only when humble

Learning is good only when humble. 2 Nephi 9: 28-29

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Learning is good only when humble

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  1. Learning is good only when humble 2 Nephi 9: 28-29 28 O that cunning plan of the evil one! O the vainness, and the frailties, and the foolishness of men! When they are learned they think they are wise, and they hearken not unto the counsel of God, for they set it aside, supposing they know of themselves, wherefore, their wisdom is foolishness and it profiteth them not. And they shall perish. 29 But to be learned is good if they hearken unto the counsels of God. Discussion #7 – Node and Mesh Methods

  2. Lecture 7 – Network Analysis

    Node Voltage and Mesh Current Methods Discussion #7 – Node and Mesh Methods
  3. Network Analysis Determining the unknown branch currents and node voltages Important to clearly define all relevant variables Construct concise set of equations There are methods to follow in order to create these equations This is the subject of the next few lectures Discussion #7 – Node and Mesh Methods
  4. Network Analysis Network Analysis Methods: Node voltage method Mesh current method Superposition Equivalent circuits Source transformation Thévenin equivalent Norton equivalent Discussion #7 – Node and Mesh Methods

  5. Node Voltage Method

    Network Analysis Discussion #7 – Node and Mesh Methods
  6. + v3 – + v1 – c b a R1 R3 + v2 – + v4 – + ia ic _ vs R4 R2 ib d Node Voltage Method Identify all node and branch voltages Discussion #7 – Node and Mesh Methods
  7. vb + R3 – + R1 – va vd i3 + R2 – i1 i va i2 vb vc R Node Voltage Method The most general method for electrical circuit analysis Based on defining the voltage at each node One node is selected as a reference node (often ground) All other voltages given relative to reference node n – 1 equations of n – 1 independent variables (node voltages) Once node voltages are determined, Ohm’s law can determine branch currents Branch currents are expressed in terms of one or more node voltages Discussion #7 – Node and Mesh Methods
  8. Node Voltage Method Label all currents and voltages (choose arbitrary orientations unless orientations are already given) Select a reference node (usually ground) This node usually has most elements tied to it Define the remaining n – 1 node voltages as independent or dependent variables Each of the mvoltage sources is associated with a dependent variable If a node is not connected to a voltage source then its voltage is treated as an independent variable Apply KCL at each node labeled as an independent variable Express currents in terms of node voltages Solve the linear system of n – 1 – m unknowns Discussion #7 – Node and Mesh Methods
  9. + R2 – + R3 – + R1 – is Node Voltage Method Example1: find expressions for each of the node voltages and the currents Discussion #7 – Node and Mesh Methods
  10. Node a Node b + R2 – i1 i2 i3 + R3 – + R1 – is Node c Node Voltage Method Example1: find expressions for each of the node voltages and the currents Label currents and voltages (polarities “arbitrarily” chosen) Choose Node c (vc) as the reference node (vc = 0) Define remaining n – 1 (2) voltages va is independent since it is not associated with a voltage source vb is independent Apply KCL at nodes a and b (node c is not independent) to find expressions for i1, i2, i3 va vb vc Discussion #7 – Node and Mesh Methods
  11. Node a Node b + R2 – i1 i2 i3 + R3 – + R1 – is Node c Node Voltage Method Example1: find expressions for each of the node voltages and the currents Apply KCL to find expressions for i1, i2, i3 va vb NB: whenever a node connects only 2 branches the same current flows in the 2 branches (EX: i2 = i3) vc Discussion #7 – Node and Mesh Methods
  12. Node a Node b + R2 – i1 i2 i3 + R3 – + R1 – is Node c Node Voltage Method Example1: find expressions for each of the node voltages and the currents Express currents in terms of node voltages va vb vc Discussion #7 – Node and Mesh Methods
  13. Node a Node b + R2 – i1 i2 i3 + R3 – + R1 – is Node c Node Voltage Method Example1: find expressions for each of the node voltages and the currents Solve the n – 1 – m equations va vb vc Discussion #7 – Node and Mesh Methods
  14. R3 R2 I1 R4 I2 R1 Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Discussion #7 – Node and Mesh Methods
  15. R3 + R3 – i3 R2 + R2 – Node a Node b i2 i1 + I1 – i4 + R4 – + R1 – I1 – I2 + R4 I2 R1 Node c Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Discussion #7 – Node and Mesh Methods
  16. Node a vb + R3 – Node b va i3 + R2 – i2 + I1 – – I2 + + R4 – + R1 – i4 i1 vc Node c Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Label currents and voltages (polarities “arbitrarily” chosen) Choose Node c (vc) as the reference node (vc = 0) Define remaining n – 1 (2) voltages va is independent vb is independent Apply KCL at nodes a and b Discussion #7 – Node and Mesh Methods
  17. Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Node a vb + R3 – Node b va i3 Apply KCL at nodes a and b + R2 – i2 + I1 – – I2 + + R4 – + R1 – i4 i1 vc Node c Discussion #7 – Node and Mesh Methods
  18. Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Node a vb + R3 – Node b va i3 Express currents in terms of voltages + R2 – i2 + I1 – – I2 + + R4 – + R1 – i4 i1 vc Node c Discussion #7 – Node and Mesh Methods
  19. Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Node a vb + R3 – Node b va i3 Solve the n – 1 – m equations + R2 – i2 + I1 – – I2 + + R4 – + R1 – i4 i1 vc Node c Discussion #7 – Node and Mesh Methods
  20. Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Node a vb + R3 – Node b va i3 Solve the n – 1 – m equations + R2 – i2 + I1 – – I2 + + R4 – + R1 – i4 i1 vc Node c Discussion #7 – Node and Mesh Methods
  21. Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Node a vb + R3 – Node b va i3 Solve the n – 1 – m equations + R2 – i2 + I1 – – I2 + + R4 – + R1 – i4 i1 vc Node c Discussion #7 – Node and Mesh Methods
  22. vb + R3 – Node b va i3 + R2 – i2 + I1 – – I2 + + R4 – + R1 – i4 i1 vc Node c Node Voltage Method Example2: solve for all unknown currents and voltages I1 = 10mA, I2 = 50mA, R1 = 1kΩ, R2= 2kΩ, R3 = 10kΩ, R4 = 2kΩ Node a Solve the n – 1 – m equations Discussion #7 – Node and Mesh Methods
  23. + – Node Voltage Method Example5: find all node voltages vs = 2V, is = 3A, R1= 1Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω R1 R3 is R4 R2 vs Discussion #7 – Node and Mesh Methods
  24. i3 i1 vc vb va + R1 – + R3 – + is – + – + R4 – + R2 – vs i2 i4 vd Node Voltage Method Example5: find all node voltages vs = 2V, is = 3A, R1= 1Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Label currents and voltages (polarities “arbitrarily” chosen) Choose Node d (vd) as the reference node (vd = 0) Define remaining n – 1 (3) voltages va is dependent (va = vs) vb is independent vc is independent Apply KCL at nodes b, and c Discussion #7 – Node and Mesh Methods
  25. i3 i1 vc vb va + R1 – + R3 – + is – + – + R4 – + R2 – vs i2 i4 vd Node Voltage Method Example5: find all node voltages vs = 2V, is = 3A, R1= 1Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Apply KCL at nodes b, and c Discussion #7 – Node and Mesh Methods
  26. i3 i1 vc vb va + R1 – + R3 – + is – + – + R4 – + R2 – vs i2 i4 vd Node Voltage Method Example5: find all node voltages vs = 2V, is = 3A, R1= 1Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Express currents in terms of voltages Discussion #7 – Node and Mesh Methods
  27. i3 i1 vc vb va + R1 – + R3 – + is – + – + R4 – + R2 – vs i2 i4 vd Node Voltage Method Example5: find all node voltages vs = 2V, is = 3A, R1= 1Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Solve the n – 1 – m equations Discussion #7 – Node and Mesh Methods
  28. R3 Vs R1 + – R2 R4 iv Node Voltage Method Example6:find the current iv Vs = 3V, Is = 2A, R1= 2Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Is Discussion #7 – Node and Mesh Methods
  29. + R3 – i3 Vs va vc + R1 – vb + – + Is – i1 + R2 – + R4 – iv i2 i4 vd Node Voltage Method Example6:find the current iv Vs = 3V, Is = 2A, R1= 2Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Label currents and voltages (polarities “arbitrarily” chosen) Choose Node d (vd) as the reference node (vd = 0) Define remaining n – 1 (3) voltages va is independent vb is dependent (actually both vb and vc are dependent on each other so choose one to be dependent and one to be independent) (vb = vc + Vs) vc is independent Apply KCL at nodes a, and c Discussion #7 – Node and Mesh Methods
  30. + R3 – i3 Vs va vc + R1 – vb + – + Is – i1 + R2 – + R4 – iv i2 i4 vd Node Voltage Method Example6:find the current iv Vs = 3V, Is = 2A, R1= 2Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Apply KCL at nodes a, and c Discussion #7 – Node and Mesh Methods
  31. + R3 – i3 Vs va vc + R1 – vb + – + Is – i1 + R2 – + R4 – iv i2 i4 vd Node Voltage Method Example6:find the current iv Vs = 3V, Is = 2A, R1= 2Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Express currents in terms of voltages Discussion #7 – Node and Mesh Methods
  32. + R3 – i3 Vs va vc + R1 – vb + – + Is – i1 + R2 – + R4 – iv i2 i4 vd Node Voltage Method Example6:find the current iv Vs = 3V, Is = 2A, R1= 2Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω Solve the n – 1 – m equations Discussion #7 – Node and Mesh Methods
  33. + – Node Voltage Method Example6:find the current iv Vs = 3V, Is = 2A, R1= 2Ω, R2= 4Ω, R3 = 2Ω, R4 = 3Ω + R3 – Solve the n – 1 – m equations i3 Vs va vc + R1 – vb + Is – i1 + R2 – + R4 – iv i2 i4 vd Discussion #7 – Node and Mesh Methods

  34. Mesh Current Method

    Network Analysis Discussion #7 – Node and Mesh Methods
  35. i i + v _ + v _ R R Mesh Current Method Write n equations of n unknowns in terms of mesh currents (where n is the number of meshes) Use of KVL to solve unknown currents Important to be consistent with current direction NB: the direction of current defines the polarity of the voltage Positive voltage Negative voltage Discussion #7 – Node and Mesh Methods
  36. + v1 – + v3 – Two meshes n = 2 R1 R3 + v2 – + v4 – + _ vs ib ia R4 R2 Mesh Current Method Write n equations of n unknowns in terms of mesh currents (where n is the number of meshes) Use of KVL to solve unknown currents Important to be consistent with current direction Discussion #7 – Node and Mesh Methods
  37. + v1 – + v3 – Two meshes n = 2 R1 R3 + v2 – + v4 – + _ vs ib ia R4 R2 Mesh Current Method Write n equations of n unknowns in terms of mesh currents (where n is the number of meshes) Use of KVL to solve unknown currents Important to be consistent with current direction i1 (current through R1) = ia but what about i2? According to the current directions of ia and ib, and the polarity of v2: i2(current through R2) = ia – ib Discussion #7 – Node and Mesh Methods
  38. + v1 – + v3 – R1 R3 + v2 – + v4 – + _ vs ib ia R4 R2 Mesh Current Method Write n equations of n unknowns in terms of mesh currents (where n is the number of meshes) Use of KVL to solve unknown currents Important to be consistent with current direction Discussion #7 – Node and Mesh Methods
  39. + _ vs Mesh Current Method Write n equations of n unknowns in terms of mesh currents (where n is the number of meshes) Use of KVL to solve unknown currents Important to be consistent with current direction + v1 – + v3 – R1 R3 + v2 – + v4 – ib ia R4 R2 Discussion #7 – Node and Mesh Methods
  40. Mesh Current Method Label each mesh current consistently Current directions are chosen arbitrarily unless given Label the voltage polarity of each circuit element Strategically (based on current direction) choose polarity unless already given In a circuit with n meshes and m current sources n – m independent equations result Each of the mcurrent sources is associated with a dependent variable If a mesh is not connected to a current source then its voltage is treated as an independent variable Apply KVL at each mesh associated with independent variables Express voltages in terms of mesh currents Solve the linear system of n – m unknowns Discussion #7 – Node and Mesh Methods
  41. R4 R1 vs1 R3 R2 vs2 + + _ _ R5 Mesh Current Method Example8: find the voltages across the resistors Vs1 = Vs2 = 110V, R1= 15Ω, R2= 40Ω, R3 = 16Ω, R4 = R5 = 1.3Ω Discussion #7 – Node and Mesh Methods
  42. + R4 – + R1 – vs1 ib + R3 – ic + R2 – vs2 + + ia _ _ – R5 + Mesh Current Method Example8: find the voltages across the resistors Vs1 = Vs2 = 110V, R1= 15Ω, R2= 40Ω, R3 = 16Ω, R4 = R5 = 1.3Ω Mesh current directions given Voltage polarities chosen and labeled Identify n – m (3) mesh currents ia is independent ia is independent ic is independent Apply KVL around meshes a, b, and c Discussion #7 – Node and Mesh Methods
  43. + R4 – + R1 – vs1 ib + R3 – ic + R2 – vs2 + + ia _ _ – R5 + Mesh Current Method Example8: find the voltages across the resistors Vs1 = Vs2 = 110V, R1= 15Ω, R2= 40Ω, R3 = 16Ω, R4 = R5 = 1.3Ω Apply KVL at nodes a,b, and c Discussion #7 – Node and Mesh Methods
  44. + R4 – + R1 – vs1 ib + R3 – ic + R2 – vs2 + + ia _ _ – R5 + Mesh Current Method Example8: find the voltages across the resistors Vs1 = Vs2 = 110V, R1= 15Ω, R2= 40Ω, R3 = 16Ω, R4 = R5 = 1.3Ω Express voltages in terms of currents Discussion #7 – Node and Mesh Methods
  45. + R4 – + R1 – vs1 ib + R3 – ic + R2 – vs2 + + ia _ _ – R5 + Mesh Current Method Example8: find the voltages across the resistors Vs1 = Vs2 = 110V, R1= 15Ω, R2= 40Ω, R3 = 16Ω, R4 = R5 = 1.3Ω Solve the n – m equations Discussion #7 – Node and Mesh Methods
  46. + R4 – + R1 – vs1 ib + R3 – ic + R2 – vs2 + + ia _ _ – R5 + Mesh Current Method Example8: find the voltages across the resistors Vs1 = Vs2 = 110V, R1= 15Ω, R2= 40Ω, R3 = 16Ω, R4 = R5 = 1.3Ω Solve the n – m equations Discussion #7 – Node and Mesh Methods
  47. R4 ic R3 R1 R2 vs Is ia ib – + Mesh Current Method Example9: find the mesh currents Vs = 6V, Is = 0.5A, R1= 3Ω, R2= 8Ω, R3 = 6Ω, R4 = 4Ω Discussion #7 – Node and Mesh Methods
  48. – + Mesh Current Method Example9: find the mesh currents Vs = 6V, Is = 0.5A, R1= 3Ω, R2= 8Ω, R3 = 6Ω, R4 = 4Ω +R4– Mesh current directions given Voltage polarities chosen and labeled Identify n – m (3) mesh currents ia is dependent (ia = is) ia is independent ic is independent Apply KVL around meshes b and c ic +R3 – + R1– + R2 – vs Is ia ib Discussion #7 – Node and Mesh Methods
  49. – + Mesh Current Method Example9: find the mesh currents Vs = 6V, Is = 0.5A, R1= 3Ω, R2= 8Ω, R3 = 6Ω, R4 = 4Ω +R4– Apply KVL at nodes b and c ic +R3 – + R1– + R2 – vs Is ia ib Discussion #7 – Node and Mesh Methods
  50. +R4– ic +R3 – + R1– + R2 – vs Is ia ib – + Mesh Current Method Example9: find the mesh currents Vs = 6V, Is = 0.5A, R1= 3Ω, R2= 8Ω, R3 = 6Ω, R4 = 4Ω Express voltages in terms of currents Discussion #7 – Node and Mesh Methods
  51. +R4– ic +R3 – + R1– + R2 – vs Is ia ib – + Mesh Current Method Example9: find the mesh currents Vs = 6V, Is = 0.5A, R1= 3Ω, R2= 8Ω, R3 = 6Ω, R4 = 4Ω Solve the n – m equations Discussion #7 – Node and Mesh Methods

  52. Equation Solver Methods

    Calculator Matrix Cramer’s Rule Brute Force (Substitution) Discussion #7 – Node and Mesh Methods
  53. TI-89 Equation Solver Press F1 Select option number 8 (Clear Home) Press APPS Select option number 1 (FlashApps) Select Simultaneous Eqn Solver Select New… In the new box enter n (the number of equations and unknowns) Discussion #7 – Node and Mesh Methods
  54. TI-89 Equation Solver Input the equation coefficients For the set of 3 equations and 3 unknowns from example 8: Input the coefficients as follows: Discussion #7 – Node and Mesh Methods
  55. TI-89 Equation Solver Press F5 (Solve) Discussion #7 – Node and Mesh Methods
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