Transcript Chapter 3
Chapter 3 Methods of Analysis Basic Theory of Circuits, SJTU 1 Cut-set Analysis Loop Analysis Analysis Methods Nodal Analysis Mesh Analysis Basic Theory of Circuits, SJTU 2 Nodal Analysis 1. 2. 3. Steps to Determine Node Voltages: Select a node as the reference node(ground), define the node voltages V1, V2,… Vn-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. Basic Theory of Circuits, SJTU 3 Fig. 3.2 Typical circuit for nodal analysis Basic Theory of Circuits, SJTU 4 So at node 1 and node 2, we can get the following equations. Basic Theory of Circuits, SJTU 5 In terms of the conductance, equations become Can also be cast in matrix form as Some examples Basic Theory of Circuits, SJTU 6 Fig. 3.5 For Example 3.2: (a) original circuit, (b) circuit for analysis Basic Theory of Circuits, SJTU 7 Nodal Analysis with Voltage Sources(1) Case 1 a voltage source is connected between the reference node and a nonreference node Basic Theory of Circuits, SJTU 8 Nodal Analysis with Voltage Sources(2) Case 2 the voltage source (dependent or independent) is connected between two nonreference nodes Basic Theory of Circuits, SJTU 9 Nodal Analysis with Voltage Sources(3) Case 3 a voltage source (dependent or independent) is connected with a resistor in series i V 11 V22 Basic Theory of Circuits, SJTU 10 Nodal Analysis with Voltage Sources(3) Example P113 Basic Theory of Circuits, SJTU 11 Mesh 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. And then you can get all voltages and currents on any branch. Basic Theory of Circuits, SJTU 12 Fig. 3.17 A circuit with two meshes Basic Theory of Circuits, SJTU 13 In matrix form: Basic Theory of Circuits, SJTU 14 Fig. 3.18 For Example 3.5 Basic Theory of Circuits, SJTU 15 Mesh Analysis with Current Sources(1) Case 1 When a current source exists only in one mesh Basic Theory of Circuits, SJTU 16 Mesh Analysis with Current Sources(2) Case 2 When a current source exists between two meshes Basic Theory of Circuits, SJTU 17 Solution 1: supermesh Solution 2: Basic Theory of Circuits, SJTU 18 Fig. 3.24 For Example 3.7 Basic Theory of Circuits, SJTU 19 Fig. 3.31 For Example 3.10 Basic Theory of Circuits, SJTU 20 Fig. 3.32 For Example 3.10; the schematic of the circuit in Fig. 3.31. Basic Theory of Circuits, SJTU 21 Nodal Versus Mesh Analysis 1. 2. 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 Basic Theory of Circuits, SJTU 22 Topological Properties Topological Concepts: 1. Graph: Assigned Graph: Subgraph: Connected Graph: Consists of redrawing the circuit, with a line representing each branch of the network. A graph with assigned branch variables. A graph G1 is said to be a subgraph of the graph G if all the nodes and branches in G1 are also in G and if each branch in G1 has the same end nodes as in G. A graph G is said to be a connected graph if between any two nodes in graph G there is at least one path. Basic Theory of Circuits, SJTU 23 Topological Properties Topological Concepts: 2. Tree: A tree T is defined as a subgraph of a connected graph G in the following conditions. 1) It must be a connected graph 2) It contains no loops 3) It includes all nodes of graph G Tree branches: the branches of a tree Cotree (complementary tree): the remaining part of a tree in graph G Links: the braches of a cotree Basic Theory of Circuits, SJTU 24 Topological Properties Topological Concepts: 3. Fundamental loop: A loop that contains one and only one link. Basic Theory of Circuits, SJTU 25 Topological Properties Topological Concepts: 4. Fundamental cut set: Cut set: A minimal set of branches that, when cut, divides the graph into two parts. Fundamental cut set: A cut set that contains one and only one tree branch. Basic Theory of Circuits, SJTU 26 2 1 4 3 5 Basic Theory of Circuits, SJTU 27 2 1 3 5 Basic Theory of Circuits, SJTU 28 Cut-set analysis: tree-branch voltages are set to be solution variables steps 1. Draw the graph and assign it 2. Select a tree 3. Define fundamental cut sets and their direction 4. Set and solve the equations 5. Get branch currents and voltages from tree-branch voltages Basic Theory of Circuits, SJTU 29 Loop analysis: steps link currents are set to be solution variables 1. Draw the graph and assign it 2. Select a tree 3. Define fundamental loops and their direction 4. Set and solve the equations 5. Get branch currents and voltages from loop currents Basic Theory of Circuits, SJTU 30