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Transcript
Basic Electrical Engineering Lecture # 10 Course Instructor: Engr. Sana Ziafat Agenda • Planner vs Non planner circuits • Node analysis Branches and Nodes Branch: elements connected end-to-end, nothing coming off in between (in series) Node: place where elements are joined—includes entire wire Nodal Analysis Kirchhoff’s current law is used to develop the method referred to as nodal analysis A node is defined as a junction of two or more branches Application of nodal analysis 1. Determine the number of nodes within the network. 2. Pick a reference node, and label each remaining node with a subscript value of voltage: V1, V2, and so on. Nodal Analysis 3. Apply Kirchhoff’s current law at each node except the reference. 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. 4. Solve the resulting equation for the nodal voltages. Nodal Analysis On occasion there will be independent voltage sources in the network to which nodal analysis is to be applied. If so, convert the voltage source to a current source (if a series resistor is present) and proceed as before or use the supernode approach: 1. Assign a nodal voltage to each independent node of the network. 2. Mentally replace independent voltage sources with short-circuits. 3. Apply KCL to the defined nodes of the network. 4. Relate the defined nodes to the independent voltage source of the network, and solve for the nodal voltages. Nodal Analysis 1. Choose a reference node and assign a subscripted voltage label to the (N – 1) remaining nodes of the network. 2. The number of equations required for a complete solution is equal to the number of subscripted voltages (N – 1). Column 1 of each equation is formed by summing the conductances tied to the node of interest and multiplying the result by that subscripted nodal voltage. Nodal Analysis (Format Approach) 3. We must now consider the mutual terms that are always subtracted form the first column. It is possible to have more than one mutual term if the nodal voltage of current interest has an element in common with more than one nodal voltage. Each mutual term is the product of the mutual conductance and the other nodal voltage tied to that conductance. Nodal Analysis (Format Approach) 4. The column to the right of the equality sign is the algebraic sum of the current sources tied to the node of interest. A current source is assigned a positive sign if it supplies current to a node and a negative sign if it draws current from the node. 5. Solve the resulting simultaneous equations for the desired voltages. Node Voltages The voltage drop from node X to a reference node (ground) is called the node voltage Vx. Example: a + Va _ b + + _ Vb _ ground Nodal Analysis Method 1. Choose a reference node ( ground, node 0) (look for the one with the most connections, or at the bottom of the circuit diagram) 2. Define unknown node voltages (those not connected to ground by voltage sources). 3. Write KCL equation at each unknown node. ▫ How? Each current involved in the KCL equation will either come from a current source (giving you the current value) or through a device like a resistor. ▫ If the current comes through a device, relate the current to the node voltages using I-V relationship (like Ohm’s law). 4. Solve the set of equations (N linear KCL equations for N unknown node voltages). Example node voltage set R1 + - V1 Va R 3 R2 What if we used different ref node? Vb R4 IS reference node • • • Choose a reference node. Define the node voltages (except reference node and the one set by the voltage source). Apply KCL at the nodes with unknown voltage. Va V1 Va Va Vb 0 R1 R2 R3 • Vb Va Vb IS R3 R4 Solve for Va and Vb in terms of circuit parameters. Example R1 Va R3 V 1 R2 I1 R4 R5 V2 Va V1 Va Va V2 I1 R1 R4 R5 Q&A