Electric Circuits I. Nodal Analysis. Dr. Firas Obeidat


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1 Electric Circuits I Nodal Analysis Dr. Firas Obeidat 1
2 Nodal Analysis Without Voltage Source Nodal analysis, which is based on a systematic application of Kirchhoff s current law (KCL). A node is defined as a junction of two or more branches. define one node of any network as a reference (that is, a point of zero potential or ground), the remaining nodes of the network will all have a fixed potential relative to this reference. For a network of N nodes, therefore, there will exist (N1) nodes with a fixed potential relative to the assigned reference node. 2
3 Nodal Analysis Without Voltage Source Steps to Determine Node Voltages 1 Determine the number of nodes within the network. 2 Select a node as the reference node. Assign voltages v 1,v 2,v 3,,v n1 to the remaining nodes. The voltages are referenced with respect to the reference node. 3 Apply KCL to each of the nonreference nodes. Use Ohm s law to express the branch currents in terms of node voltages. Assume that all unknown currents leave the node for each application of Kirchhoff s current law. In other words, for each node, don t be influenced by the direction that an unknown current for another node may have had. Each node is to be treated as a separate entity, independent of the application of Kirchhoff s current law to the other nodes. Current flows from a higher potential to a lower potential in a resistor. 3
4 Nodal Analysis Without Voltage Source Steps to Determine Node Voltages This principle can be expressed as = 4 Solve the resulting simultaneous equations to obtain the unknown node voltages. Example: Calculate the node voltages in the circuit? Node =0 (1) Multiplying each term by 4, we obtain 3 = 20 (2) Node =0 (3) 4
5 Nodal Analysis Without Voltage Source Multiplying each term by 12, we obtain = 60 (4) Using the elimination technique to solve equations (2) and (4) to get v 1 and v 2 3 = 20 (2) = 60 (4) 4 = 80 = 20 Substituting v2 in equation (2) 3 20 = 20 (2) = 40 3 =
6 Nodal Analysis Without Voltage Source Example: Calculate the node voltages in the circuit? Node =0 (1) Multiplying by 4 and rearranging terms 3 2 = 12 (2) Node =0 (3) Multiplying by 8 and rearranging terms 4 +7 = 0 (4) Node =0 (5) 6
7 Nodal Analysis Without Voltage Source = (6) = 12 5 = 2.4 (9) Substitute eq.(6) in eq.(5) 2( ) + + =0 (7) Multiplying by 8 and rearranging terms = = 0 (8) Solve equations (2), (4), and (8) using elimination technique to get v 1, v 2 and v 3 Add eq.(2) to eq.(8) 3 2 = 12 (2) = 0 (8) 5 5 = 12 Add eq.(4) to eq.(8) 4 +7 = 0 (4) = 0 (8) 2 +4 = 0 = 2 (10) Substitute eq.(10) in eq.(9) 2 = 2.4 = 2.4 ( 11) = 2 = = 4.8 (12) Substitute eq.(11) in eq.(12) in eq.(8) = 0 = 2. 4 (8) 7
8 Nodal Analysis Without Voltage Source Example: Calculate the node voltages in the circuit? Node =0 (1) = 5 (2) Node =0 (3) = 5 (4) Node =0 (5) = 12 (6) Solve eq.(2), eq.(4) and eq.(6) to get v 1, v 2, and v 3. 8
9 Nodal Analysis With Voltage Source 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. In the figure for example, v 1 =10V. CASE 2: If the voltage source (dependent or independent) is connected between two nonreference nodes, the two nonreference nodes form a generalized node or supernode; we apply both KCL and KVL to determine the node voltages. 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. 9
10 Nodal Analysis With Voltage Source In the figure shown, nodes 2 and 3 form a supernode. Applying KCL on supernode, then we get =0 Applying KVL on supernode, then we get = 5 Note the following properties of a supernode: 1. The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages. 2. A supernode has no voltage of its own. 3. A supernode requires the application of both KCL and KVL. 10
11 Nodal Analysis With Voltage Source Example: For the circuit shown, find the node voltages. The supernode contains the 2V source, nodes 1 and 2, and the 10Ω resistor. Applying KCL to the supernode gives = = 0 = 20 2 (1) Applying KVL to the supernode gives =2 or = +2 (2) From eq.(1) and eq.(2) = +2 = =22 =7.33V == +2= =5.33V Note that the 10 resistor does not make any difference because it is connected across the supernode. 11
12 Nodal Analysis With Voltage Source Example: For the circuit shown, find the node voltages. Node = = 11 (1) Supernode = = 28 (2) = 22 (3) Solve eq.(1), eq.(2) and eq.(3) to get v 1, v 2, and v 3. 12
13 Nodal Analysis With Voltage Source Example: For the circuit shown, find the node voltages. Node = 5 (1) Node 2 = 18 (2) Supernode = 12 (3) 18 = 12 = 30 Substitute v 2 and v 3 in eq.(1) = 5 =
14 Nodal Analysis With Voltage Source Example: For the circuit shown, find the node voltages. Node 1 = 12 (1) Node =0 (2) Supernode =0 (3) 0.5( ) =0 (3) = 0.2 (4) = 0.2( ) (4) Rearrange these four equations then we have the following equations = 12 (5) = 14 (6) = 0 (7) = 0 (8) = 12 = 4 = 0 = 42 14
15 15
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