eeng2009lec5 - 6.1 Meshes and Mesh Currents EENG 2009 Part...

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EENG 2009 FS 2006 Part 6: Mesh Analysis 152 EENG 2009 Part 6. Mesh Analysis 6.1 Meshes and Mesh Currents 6.2 Mesh Analysis By Example 6.3 Supermeshes 6.4 Nodal Analysis vs. Mesh Analysis EENG 2009 FS 2006 Part 6: Mesh Analysis 153 6.1 Meshes and Mesh Currents For planar networks the meshes are the “windows” formed when the network is drawn with no branches crossing. Formally, a mesh is a circuit loop that does not enclose any elements. For example: R 1 R 6 R 2 + R 5 V R 7 R 8 R 4 R 3 R 1 R 6 R 2 + R 5 V R 8 R 4 R 3 R 7 mesh currents: R 1 R 6 R 2 + R 5 V R 8 R 4 R 3 R 7 i 1 i 2 i 3 i 4
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EENG 2009 FS 2006 Part 6: Mesh Analysis 154 R 1 R 6 R 2 + R 5 V R 8 R 4 R 3 R 7 i 1 i 2 i 3 i 4 It is not always possible to identify a mesh current with a single branch current, and so it is not always possible to directly measure a mesh current: i 1 , i 2 , and i 4 are both mesh currents and branch currents and can be measured by placing an ammeter at the proper location. i 3 is a mesh current but not a branch current; it cannot be measured directly. R 2 R 8 R 4 R 3 i 2 i 3 i 1 ab c d The branch- current relationships: i ab = i 2 i bc = i 2 i 4 i cd = i 3 i 4 i da = i 3 i 1 i ac = i 3 i 2 i 4 R 6 EENG 2009 FS 2006 Part 6: Mesh Analysis 155 6.2 Mesh Analysis By Example Example 1 Find the mesh-analysis equations. R 1 R 3 R 2 + v 2 + v 1 i 1 i 3 i 2 Solution: Note that we are starting with branch currents— not mesh currents (although i 1 and i 2 are branch and also mesh currents). KCL: i 1 = i 2 + i 3 (1) KVL: R 1 i 1 + R 3 i 3 = v 1 (2) KVL: – R 2 i 2 + R 3 i 3 = v 2 (3) Use (1) to eliminate i 3 : i 3 = i 1 –i 2 , and substitute into (2) and (3) : R 1 i 1 + R 3 ( i 1 2 ) = v 1 –R 2 i 2 + R 3 ( i 1 2 ) = v 2
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EENG 2009 FS 2006 Part 6: Mesh Analysis 156 (cont.) We can get the same set of two equations (without having to eliminate the “extra” variable i 3 ) by writing KVL in terms of the mesh currents i 1 and i 2 . R 1 R 3 R 2 + v 2 + v 1 i 1 i 2 KVL1 : R 1 i 1 + R 3 ( i 1 –i 2 ) = v 1 KVL2 : R 3 ( i 2 1 ) + R 2 i 2 = – v 2 , or –R 2 i 2 + R 3 ( i 1 2 ) = v 2 These are the same two equations as the two equations developed from the branch currents: R 1 i 1 + R 3 ( i 1 2 ) = v 1 2 i 2 + R 3 ( i 1 2 ) = v 2 The rationale for mesh analysis is that once the mesh currents are determined, all the other voltages and currents can be obtained in a simple manner.
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eeng2009lec5 - 6.1 Meshes and Mesh Currents EENG 2009 Part...

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