Fundamentals of Electric Circuits

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92 Chapter 3 Methods of Analysis We now apply KVL to the branches involving the voltage sources as shown in Fig. 3.13(b). For loop 1, v 1 20 v 2 0 1 v 1 v 2 20 (3.4.3) For loop 2, v 3 3v x v 4 0 But v x v 1 v 4 so that 3v 1 v 3 2v 4 0 (3.4.4) For loop 3, v x 3v x 6i 3 20 0 But 6i 3 v 3 v 2 and v x v 1 v 4 . Hence, 2v 1 v 2 v 3 2v 4 20 (3.4.5) We need four node voltages, v 1 , v 2 , v 3 , and v 4 , and it requires only four out of the five Eqs. (3.4.1) to (3.4.5) to find them. Although the fifth equation is redundant, it can be used to check results. We can solve Eqs. (3.4.1) to (3.4.4) directly using MATLAB. We can eliminate one node voltage so that we solve three simultaneous equations instead of four. From Eq. (3.4.3), v 2 v 1 20. Substituting this into Eqs. (3.4.1) and (3.4.2), respectively, gives and 6v 1 v 3 2v 4 80 6v 1 5v 3 16v 4 40 (3.4.6) (3.4.7) Equations (3.4.4), (3.4.6), and (3.4.7) can be cast in matrix form as Using Cramer’s rule gives 3 1 2 v 1 0 £ 6 1 2 § £ v 3 § £ 80 § 6 5 16 v 4 40 3 1 2 0 1 2 ¢ † 6 1 2 † 18, ¢ 1 † 80 1 2 † 480, 6 5 16 40 5 16 3 0 2 3 1 0 ¢ 3 † 6 80 2 † 3120, ¢ 4 † 6 1 80 † 840 6 40 16 6 5 40 Thus, we arrive at the node voltages as v 1 ¢ 1 , ¢ 480 18 26.67 V, v 3 ¢ 3 ¢ 3120 18 173.33 V v 4 ¢ 4 ¢ 840 18 46.67 V and v 2 v 1 20 6.667 V. We have not used Eq. (3.4.5); it can be used to cross check results.
3.4 Mesh Analysis 93 Find v 1 , v 2 , and in the circuit of Fig. 3.14 using nodal analysis. v 3 Answer: v 1 3.043 V, v 2 6.956 V, v 3 0.6522 V. Practice Problem 3.4 6 Ω 10 V v v 5i 2 1 + − v 3 i + − 3.4 Mesh Analysis Mesh analysis provides another general procedure for analyzing circuits, using mesh currents as the circuit variables. Using mesh currents instead of element currents as circuit variables is convenient and reduces the number of equations that must be solved simultaneously. Recall that a loop is a closed path with no node passed more than once. A mesh is a loop that does not contain any other loop within it. Nodal analysis applies KCL to find unknown voltages in a given circuit, while mesh analysis applies KVL to find unknown currents. Mesh analysis is not quite as general as nodal analysis because it is only applicable to a circuit that is planar. A planar circuit is one that can be drawn in a plane with no branches crossing one another; otherwise it is nonplanar. A circuit may have crossing branches and still be planar if it can be redrawn such that it has no crossing branches. For example, the circuit in Fig. 3.15(a) has two crossing branches, but it can be redrawn as in Fig. 3.15(b). Hence, the circuit in Fig. 3.15(a) is planar. However, the circuit in Fig. 3.16 is nonplanar, because there is no way to redraw it and avoid the branches crossing. Nonplanar circuits can be handled using nodal analysis, but they will not be considered in this text. 2 Ω 4 Ω 3 Ω Figure 3.14 For Practice Prob. 3.4. Mesh analysis is also known as loop analysis or the mesh-current method. 1 Ω 1 A 2 Ω 5 Ω 6 Ω 4 Ω 3 Ω 8 Ω 7 Ω 1 Ω (a) 5 Ω 6 Ω 4 Ω 3 Ω 7 Ω 2 Ω 1 A 2 Ω 13 Ω 5 A 12 Ω 11 Ω 9 Ω 8 Ω 5 Ω 1 Ω 3 Ω 4 Ω 6 Ω Figure 3.16 A nonplanar circuit. 10 Ω To understand mesh analysis, we should first explain more about what we mean by a mesh. 8 Ω 7 Ω (b) Figure 3.15 (a) A planar circuit with crossing branches, (b) the same circuit redrawn with no crossing branches. A mesh is a loop which does not contain any other loops within it.
Page 1 and 2: c h a p t e r Methods of Analysis 3
Page 3 and 4: 3.2 Nodal Analysis 83 since it is a
Page 5 and 6: 3.2 Nodal Analysis 85 Now we have t
Page 7 and 8: 3.2 Nodal Analysis 87 Substituting
Page 9 and 10: 3.3 Nodal Analysis with Voltage Sou
Page 11: 3.3 Nodal Analysis with Voltage Sou
Page 15 and 16: 3.4 Mesh Analysis 95 The third step
Page 17 and 18: 3.4 Mesh Analysis 97 But at node A,
Page 19 and 20: 3.5 Mesh Analysis with Current Sour
Page 21 and 22: 3.6 Nodal and Mesh Analyses by Insp
Page 23 and 24: 3.6 Nodal and Mesh Analyses by Insp
Page 25 and 26: 3.8 Circuit Analysis with PSpice 10
Page 27 and 28: 3.9 Applications: DC Transistor Cir
Page 33 and 34: Review Questions 113 4. A supermesh
Page 35 and 36: Problems 115 3.9 Determine in the c
Page 37 and 38: Problems 117 3.20 For the circuit i
Page 39 and 40: Problems 119 Sections 3.4 and 3.5 M
Page 41 and 42: Problems 121 3.47 Rework Prob. 3.19
Page 43 and 44: Problems 123 i 1 i 2 3.62 Find the
Page 45 and 46: Problems 125 3.77 Solve for V 1 and

Fundamentals of Electric Circuits

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