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CIRCUIT ANALYSIS METHODS

CIRCUIT ANALYSIS METHODS. Topic 3. Motivation (1). If you are given the following circuit, how can we determine (1) the voltage across each resistor, (2) current through each resistor. (3) power generated by each current source, etc.

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CIRCUIT ANALYSIS METHODS

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  1. CIRCUIT ANALYSIS METHODS Topic 3

  2. Motivation (1) If you are given the following circuit, how can we determine (1) the voltage across each resistor, (2) current through each resistor. (3) power generated by each current source, etc. What are the things which we need to know in order to determine the answers?

  3. Motivation (2) Things we need to know in solving any resistive circuit with current and voltage sources only: • Kirchhoff’s Current Laws (KCL) • Kirchhoff’s Voltage Laws (KVL) • Ohm’s Law How should we apply these laws to determine the answers?

  4. There are four ways of solving simultaneous equations: • Cramer’s rule • Calculator (real numbers only) • Normal substitution and elimination (not more than two equations) • Computer program packages: mathcad, maple, mathematica etc.

  5. CIRCUIT ANALYSIS METHODS • Node-Voltage method • Mesh-current method • Source transformation • Thevenin equivalent circuit • Norton equivalent circuit • Maximum power transfer • Superposition principle

  6. INTRODUCTION OF NODE-VOLTAGE METHOD (NODAL ANALYSIS) • In nodal analysis, we are interested in finding the node voltage. • Assume that circuits do no contain voltage sources. • Use KCL. • Important step: select one of the node as reference node • Then define the node voltage in the circuit diagram.

  7. NODAL ANALYSIS -V1+ -V2+ -V1+ -V2+ STEP 1: Select a node as reference node and assign V1 and V2 to the remaining nodes STEP 2:Apply KCL at each node in the circuit. Use Ohm’s Law to express the branch currents in term of node voltage. STEP 3: Solve the resulting simultaneous equation to obtain unknown node voltage **If we need the currents, we can calculate from the values of nodal voltage

  8. At node 1, applying KCL: At node 2, applying KCL: -V1+ -V2+ Applying Ohm’s Law to express the unknown current i1, i2 and i3 in term of node voltage

  9. -V1+ -V2+ Substituting in equation at node 1

  10. Substituting in equation at node 2

  11. Nodal voltage example 1 At node 1, applying KCL: Substituting in equation at node 1

  12. Nodal voltage example 1

  13. Applying Nodal Analysis on Circuit with Voltage Sources Three different effects depending on placement of voltage source in the circuit. Does the presence of a voltage source complicate or simplify the analysis?

  14. Case 1: Voltage source between two non-reference essential nodes. Supernode Equation:

  15. Case 2: Voltage source between a reference essential node and a non-reference essential node. Known node voltage:

  16. Case 3: Voltage source between an essential node and a non-essential node. Node voltage at non-essential node:

  17. Node-voltage example 2 Obtain V1 and V2

  18. In the diagram, node 3 is define as reference node and node 1 and 2 as node voltage V1 and V2. • The node-voltage equation for node 1 is,

  19. Rearrange the node – voltage equation for node 1

  20. The node-voltage • equation for node 2 is, Rearrange the node – voltage equation for node 2

  21. Solving for V1 and V2 yields

  22. EXERCISE Obtain the node voltage in the circuit Ans: V1= -2V V2= -14V

  23. THE NODE-VOLTAGE METHOD AND DEPENDENT SOURCES • If the circuit contains dependent sources, the node-voltage equations must be supplemented with the constraint equation imposed by the presence of the dependent sources.

  24. example… Use the node-voltage method to find the power dissipated in the 5Ω resistor.

  25. solution… • The circuit has 3 node. • Thus there must be 2 node-voltage equation. • Summing the currents away from node 1 generates the equation, LOOK CAREFULLY

  26. solution… • Summing the current at node 2 yields,

  27. As written, these two equations contain three unknowns namely V1, V2 and iØ. • To eliminate iØ, express the current in terms of node-voltage,

  28. Substituting this relationship into the node 2 equation,

  29. Solving for V1 and V2 gives,

  30. Then,

  31. EXERCISE Find the voltage at the three nonreference nodes in the circuit below Ans: V1=80V V2=-64V V3=156V

  32. SPECIAL CASE • When a voltage source is the only element between two essential nodes, the node-voltage method is simplified.

  33. Example…

  34. There is three essential nodes, so two simultaneous equation are needed. • Only one unknown node voltage, V2 where as V1=100V. • Therefore, only a single node-voltage equation is needed which is at node 2.

  35. Using V1 =100V, thus V2=125V.

  36. SUPERNODE • When a voltage source is between two essential nodes, those nodes can be combine to form a supernode (voltage source is assume as open circuit). • Apply both KCL and KVL to determine the node-voltage

  37. Supernode • Node 2 and 3 form a supernode • Step to determine node voltage: • Apply KCL at supernode • Apply KVL at supernode • Solve resulting simultaneous equation

  38. KCL at supernode:

  39. KVL at supernode: To apply KVL at supernode, reduce the circuit.

  40. Supernode Example 1 Determine the nodal voltages in Figure using the concept of a super node.

  41. Solution • KCL at supernode

  42. Solution • KVL at supernode

  43. Supernode example 2

  44. Nodes chosen,

  45. Node-voltage equation for node 2 and 3,

  46. Summing both equation, Above equation can be generates directly using supernode approach

  47. Equation at node 2 and 3; • From circuit V1=50V

  48. Supernod

  49. Starting with resistor 50Ω branch and moving counterclockwise around the supernode,

  50. Using V1 =50V and V3 as a function of V2,

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