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

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  1. Schedule… Discussion #7 – Node and Mesh Methods

  2. 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

  3. Lecture 7 – Network Analysis Node Voltage and Mesh Current Methods Discussion #7 – Node and Mesh Methods

  4. 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

  5. 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

  6. Node Voltage Method Network Analysis Discussion #7 – Node and Mesh Methods

  7. + 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

  8. 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

  9. 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

  10. + 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

  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 • 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

  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 • 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

  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 • Express currents in terms of node voltages va vb vc Discussion #7 – Node and Mesh Methods

  14. 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

  15. 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

  16. 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

  17. 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

  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 • Apply KCL at nodes a and b + 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 • Express currents in terms of voltages + 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. 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

  23. 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

  24. + – 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

  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 • 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

  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 • Apply KCL at nodes b, and c 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 • Express currents in terms of voltages Discussion #7 – Node and Mesh Methods

  28. 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

  29. 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

  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 • 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

  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 • Apply KCL at nodes a, and c 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 • Express currents in terms of voltages Discussion #7 – Node and Mesh Methods

  33. + 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

  34. + – 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

  35. Mesh Current Method Network Analysis Discussion #7 – Node and Mesh Methods

  36. 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

  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 Discussion #7 – Node and Mesh Methods

  38. + 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

  39. + 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

  40. + _ 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

  41. 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

  42. 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

  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 • 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

  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 • Apply KVL at nodes a,b, and c 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 • Express voltages in terms of currents 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 – + 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

  48. 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

  49. – + 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

  50. – + 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

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