Once we know the nodal voltages we can find anything in the circuit eg voltage

# Once we know the nodal voltages we can find anything

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Once we know the nodal voltages, we can find anything in the circuit! e.g., voltage across the 5Ω resistor in the middle is equal to V 1 V 2 ; voltage across the 3A source is V 1 ; voltage across the 2A source is V 2 ; and currents can be found via Ohm’s law. Prof. C.K. Tse: Circuit Analysis Review
45 Nodal analysis In general, we formulate the solution in terms of unknown nodal voltages: [ G ] [ V ] = [ I ] nodal equation where [ G ] is the conductance matrix [ V ] is the unknown node voltage vector [ I ] is the source vector For a short cut in setting up the above matrix equation, see Sec. 3.3.1.2 of the textbook . This may be picked up in the tutorial. Prof. C.K. Tse: Circuit Analysis Review
46 Nodal analysis — observing superposition Consider the previous example. The nodal equation is given by: Thus, the solution can be written as Remember what 3 and 2 are? They are the sources! The above solution can also be written as or SUPERPOSITION of two sources Prof. C.K. Tse: Circuit Analysis Review
47 Problem with voltage sources The nodal method may run into trouble if the circuit has voltage source(s). Suppose we define the unknowns in the same way, i.e., V 1 , V 2 and V 3 . The trouble is that we don’t know what current is flowing through the 2V source! How can we set up the KCL equation for nodes 2 and 3? One solution is to ignore nodes 1 and 3. Instead we look at the supernode merging 2 and 3. So, we set up KCL equations for node 1 and the supernode: One more equation: V 3 V 2 = 2 Finally, solve the equations. Prof. C.K. Tse: Circuit Analysis Review
48 In all cases, we see that the mesh method ends up with N equations and N unknowns, where N is the number of nodes of the circuit minus 1. One important point: The nodal method is over-complex when applied to circuits with voltage source(s). WHY? We don’t need N equations for circuits with voltage source(s) because the node voltages are partly known! In the previous example, it seems unnecessary to solve for both V 2 and V 3 because their difference is known to be 2! This is a waste of efforts! Can we improve it? Complexity of nodal method Prof. C.K. Tse: Circuit Analysis Review
49 Final note on superposition Superposition is a consequence of linearity. We may conclude that for any linear circuit, any voltage or current can be written as linear combination of the sources. Suppose we have a circuit which contains two voltage sources V 1 , V 2 and I 3 . And, suppose we wish to find I x . Without doing anything, we know for sure that the following is correct: I x = a V 1 + b V 2 + c I 3 where a, b and c are some constants. Is this property useful? Can we use this property for analysis? We may pick this up in the tutorial. I x V 1 V 2 I 3 Prof. C.K. Tse: Circuit Analysis Review

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• Summer '16
• Martin Chow
• Thévenin's theorem, Voltage source, Norton's theorem, Current Source, Series and parallel circuits, Voltage drop

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