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Methods of Analysis Resistance

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Methods of Analysis Resistance UNIT I – BASIC CIRCUIT CONCEPTS – Circuit elements – Kirchhoff’s Law – V-I Relationship of R,L and C – Independent and Dependent sources – Simple Resistive circuits – Networks reduction – Voltage division – current source transformation. - Analysis of circuit using mesh current and nodal voltage methods. 1 Methods of Analysis Resistance 2 Methods of Analysis Ohm’s Law 3 Methods of Analysis Resistors & Passive Sign Convention 4 Methods of Analysis Example: Ohm’s Law 5 Methods of Analysis Short Circuit as Zero Resistance 6 Methods of Analysis Short Circuit as Voltage Source (0V) 7 Methods of Analysis Open Circuit 8 Methods of Analysis Open Circuit as Current Source (0 A) 9 Methods of Analysis Conductance 10 Methods of Analysis Circuit Building Blocks 11 Methods of Analysis Branches 12 Methods of Analysis Nodes 13 Methods of Analysis Loops 14 Methods of Analysis Overview of Kirchhoff’s Laws 15 Methods of Analysis Kirchhoff’s Current Law 16 Methods of Analysis Kirchhoff’s Current Law for Boundaries 17 Methods of Analysis KCL - Example 18 Methods of Analysis Ideal Current Sources: Series 19 Methods of Analysis Kirchhoff’s Voltage Law - KVL 20 Methods of Analysis KVL - Example 21 Methods of Analysis Example – Applying the Basic Laws 22 Methods of Analysis Example – Applying the Basic Laws 23 Methods of Analysis Example – Applying the Basic Laws 24 Methods of Analysis Resistors in Series 25 Methods of Analysis Resistors in Parallel 26 Methods of Analysis Resistors in Parallel 27 Methods of Analysis Voltage Divider 28 Methods of Analysis Current Divider 29 Methods of Analysis Resistor Network 30 Methods of Analysis Resistor Network - Comments 31 Methods of Analysis Delta Wye Transformations 32 Methods of Analysis Delta Wye Transformations 33 Methods of Analysis Example – Delta Wye Transformations 34 Methods of Analysis 35 Methods of Analysis Methods of Analysis • Introduction • Nodal analysis • Nodal analysis with voltage source • Mesh analysis • Mesh analysis with current source • Nodal and mesh analyses by inspection • Nodal versus mesh analysis 36 Methods of Analysis 3.2 Nodal Analysis Steps to Determine Node Voltages: 1. Select a node as the reference node. Assign voltage v1, v2, …vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node. 2. Apply KCL to each of the n-1 nonreference nodes. Use Ohm’s law to express the branch currents in terms of node voltages. 3. Solve the resulting simultaneous equations to obtain the unknown node voltages. 37 Methods of Analysis Figure 3.1 Common symbols for indicating a reference node, (a) common ground, (b) ground, (c) chassis. 38 Methods of Analysis Figure 3.2 Typical circuit for nodal analysis 39 Methods of Analysis I 1 I 2 i1 i2 I 2 i2 i3 v v i higher lower R v1 0 i1 or i1 G1v1 R1 v1 v2 i2 or i2 G2 (v1 v2 ) R2 v2 0 i3 or i3 G3v2 R3 40 Methods of Analysis v1 v1 v2 I1 I2 R1 R2 v1 v2 v2 I2 R2 R3 I1 I 2 G1v1 G2 (v1 v2 ) I 2 G2 (v1 v2 ) G3v2 G G G v I I 1 2 2 1 1 2 G2 G2 G3 v2 I2 41 Methods of Analysis Example 3.1 Calculus the node voltage in the circuit shown in Fig. 3.3(a) 42 Methods of Analysis Example 3.1 At node 1 i1 i2 i3 v v v 0 5 1 2 1 4 2 43 Methods of Analysis Example 3.1 At node 2 i2 i4 i1 i5 v v v 0 5 2 1 2 4 6 44 Methods of Analysis Example 3.1 In matrix form: 1 1 1 2 4 4 v1 5 1 1 1 v 5 2 4 6 4 45 Methods of Analysis Practice Problem 3.1 Fig 3.4 46 Methods of Analysis Example 3.2 Determine the voltage at the nodes in Fig. 3.5(a) 47 Methods of Analysis Example 3.2 At node 1, 3 i1 ix v v v v 3 1 3 1 2 4 2 48 Methods of Analysis Example 3.2 At nodeix i2 2i3 v v v v v 0 1 2 2 3 2 2 8 4 49 Methods of Analysis Example 3.2 At node 3 i1 i2 2ix v v v v 2(v v ) 1 3 2 3 1 2 4 8 2 50 Methods of Analysis Example 3.2 In matrix form: 3 1 1 4 2 4 v 3 1 1 7 1 v2 0 2 8 8 3 9 3 v3 0 4 8 8 51 Methods of Analysis 3.3 Nodal Analysis with Voltage Sources Case 1: The voltage source is connected between a nonreference node and the reference node: The nonreference node voltage is equal to the magnitude of voltage source and the number of unknown nonreference nodes is reduced by one. Case 2: The voltage source is connected between two nonreferenced nodes: a generalized node (supernode) is formed. 52 Methods of Analysis 3.3 Nodal Analysis with Voltage Sources i1 i4 i2 i3 v v v v v 0 v 0 1 2 1 3 2 3 2 4 8 6 v2 v3 5 Fig. 3.7 A circuit with a supernode. 53 Methods of Analysis A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it. The required two equations for regulating the two nonreference node voltages are obtained by the KCL of the supernode and the relationship of node voltages due to the 54 voltage source. Methods of Analysis Example 3.3 For the circuit shown in Fig. 3.9, find the node voltages. 2 7 i1i2 0 v v 2 7 1 2 0 2 4 v1 v2 2 i1 i2 55 Methods of Analysis Example 3.4 Find the node voltages in the circuit of Fig. 3.12. 56 Methods of Analysis Example 3.4 At suopernode 1-2, v v v v v 3 2 10 1 4 1 6 3 2 v1 v2 20 57 Methods of Analysis Example 3.4 At supernode 3-4, v v v v v v 1 4 3 2 4 3 3 6 1 4 v3 v4 3(v1 v4 ) 58 Methods of Analysis 3.4 Mesh Analysis Mesh analysis: another procedure for analyzing circuits, applicable to planar circuit. A Mesh is a loop which does not contain any other loops within it 59 Methods of Analysis Fig. 3.15 (a) A Planar circuit with crossing branches, (b) The same circuit redrawn with no crossing branches. 60 Methods of Analysis Fig. 3.16 A nonplanar circuit. 61 Methods of Analysis Steps to Determine Mesh Currents: 1. Assign mesh currents i1, i2, .., in to the n meshes. 2. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents. 3. Solve the resulting n simultaneous equations to get the mesh currents. 62 Methods of Analysis Fig. 3.17 A circuit with two meshes. 63 Methods of Analysis V1 R1i1 R3 (i1 i2 ) 0 Apply KVL to each mesh. For mesh 1, (R1 R3 )i1 R3i2 V1 R i V R (i i ) 0 For mesh 22,2 2 3 2 1 R3i1 (R2 R3 )i2 V2 64 Methods of Analysis Solve for the mesh currents. R R R i V 1 3 3 1 1 R3 R2 R3 i2 V2 Use i for a mesh current and I for a branch current. It’s evident from Fig. 3.17 that I1 i1, I2 i2 , I3 i1 i2 65 Methods of Analysis Example 3.5 Find the branch current I1, I2, and I3 using mesh analysis. 66 Methods of Analysis Example 3.5 15 5i1 10(i1 i2 ) 10 0 For mesh 1, 3i1 2i2 1 6i 4i 10(i i ) 10 0 For mesh2 2, 2 2 1 i1 2i2 1 I1 i1, I2 i2 , I3 i1 i2 We can find i1 and i2 by substitution method or Cramer’s rule. Then, 67 Methods of Analysis Example 3.6 Use mesh analysis to find the current I0 in the circuit of Fig. 3.20. 68 Methods of Analysis Example 3.6 Apply KVL to each mesh. For mesh 1, 24 10(i1 i2 ) 12(i1 i3 ) 0 11i1 5i2 6i3 12 For mesh 2, 24i2 4(i2 i3 ) 10(i2 i1) 0 5i1 19i2 2i3 0 69 Methods of Analysis Example 3.6 For mesh 3, 4 I 0 12 ( i 3 i1 ) 4 ( i 3 i 2 ) 0 At node A, I 0 I 1 i 2 , 4 ( i1 i 2 ) 12 ( i 3 i1 ) 4 ( i 3 i 2 ) 0 i1 i 2 2 i 3 0 In matrix from Eqs. (3.6.1) to (3.6.3) become 11 5 6 i1 12 5 19 2 i2 0 1 1 2 i3 0 we can calculus i1, i2 and i3 by Cramer’s rule, and find I0. 70 Methods of Analysis 3.5 Mesh Analysis with Current Sources Fig. 3.22 A circuit with a current source. 71 Methods of Analysis Case 1 i1 2 A – Current source exist only in one mesh – One mesh variable is reduced Case 2 – Current source exists between two meshes, a super-mesh is obtained. 72 Methods of Analysis Fig. 3.23 a supermesh results when two meshes have a (dependent , independent) current source in common. 73 Methods of Analysis Properties of a Supermesh 1. The current is not completely ignored – provides the constraint equation necessary to solve for the mesh current.
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