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Chapter 3

Chapter 3. Methods of Analysis.

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Chapter 3

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  1. Chapter 3 Methods of Analysis SJTU

  2. So far, we have analyzed relatively simple circuits by applying Kirchhoff’s laws in combination with Ohm’s law. We can use this approach for all circuits, but as they become structurally more complicated and involve more and more elements, this direct method soon becomes cumbersome. In this chapter we introduce two powerful techniques of circuit analysis: Nodal Analysis and Mesh Analysis. These techniques give us two systematic methods of describing circuits with the minimum number of simultaneous equations. With them we can analyze almost any circuit by to obtain the required values of current or voltage. SJTU

  3. Nodal Analysis • Steps to Determine Node Voltages: • Select a node as the reference node(ground), define the node voltages 1, 2,… n-1 to the remaining n-1nodes . The voltages are referenced with respect to the reference node. • Apply KCL to each of the n-1 independent nodes. Use Ohm’s law to express the branch currents in terms of node voltages. • Solve the resulting simultaneous equations to obtain the unknown node voltages. SJTU

  4. Fig. 3.2 Typical circuit for nodal analysis SJTU

  5. So at node 1 and node 2, we can get the following equations. SJTU

  6. In terms of the conductance, equations become Can also be cast in matrix form as Some examples SJTU

  7. Fig. 3.5 For Example 3.2: (a) original circuit, (b) circuit for analysis SJTU

  8. Nodal Analysis with Voltage Sources(1) • Case 1 If a voltage source is connected between the reference node and a nonreference node, we simply set the voltage at the nonreference node equal to the voltage of the voltage source. As in the figure right: SJTU

  9. Nodal Analysis with Voltage Sources(2) • Case 2 If the voltage source (dependent or independent) is connected between two nonreference nodes, the two nonreference nodes form a supernode; we apply both KVL and KCL to determine the node voltages. As in the figure right: SJTU

  10. Nodal Analysis with Voltage Sources(3) • Case 3 If a voltage source (dependent or independent) is connected with a resistor in series, we treat them as one branch. As in the figure right: i V11 V22 SJTU

  11. Nodal Analysis with Voltage Sources(3) Example P113 SJTU

  12. Mesh Analysis • Steps to Determine Mesh Currents: • Assign mesh currents i1, i2,…in to the n meshes. • Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents. • Solve the resulting n simultaneous equations to get the mesh currents. SJTU

  13. Fig. 3.17 A circuit with two meshes SJTU

  14. In matrix form: SJTU

  15. Fig. 3.18 For Example 3.5 SJTU

  16. Mesh Analysis with Current Sources(1) • Case 1 When a current source exists only in one mesh: Consider the figure right. SJTU

  17. Mesh Analysis with Current Sources(2) • Case 2 When a current source exists between two meshes:Consider the figure right. 2 solutions: • Set v as the voltage across the current source, then add a constraint equation. • Use supermesh to solve the problem. SJTU

  18. Solution 1: supermesh Solution 2: Note: 1, The current source in the supermesh is not completely ignored; it provides the constraint equation necessary to solve for the mesh current. 2, A supermesh has no current of its own. 3, A supermesh requires the application of both KVL and KCL. SJTU

  19. Fig. 3.24 For Example 3.7 supermesh SJTU

  20. Fig. 3.31 For Example 3.10 SJTU

  21. Fig. 3.32 For Example 3.10; the schematic of the circuit in Fig. 3.31. SJTU

  22. Nodal Versus Mesh Analysis • Both provide a systematic way of analyzing a complex network. • When is the nodal method preferred to the mesh method? • A circuit with fewer nodes than meshes is better analyzed using nodal analysis, while a circuit with fewer meshes than nodes is better analyzed using mesh analysis. • Based on the information required. Node voltages required--------nodal analysis Branch or mesh currents required-------mesh analysis SJTU

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