The co tree will have all the current sources Prof CK Tse Graph Theory

The co tree will have all the current sources prof ck

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The co-tree will have all the current sources.
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Prof. C.K. Tse: Graph Theory & Systematic Analysis 29 Standard tree number of nodes n = 4 number of branches b = 5 number of tree branches t = n –1 = 3
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Prof. C.K. Tse: Graph Theory & Systematic Analysis 30 Two systematic approaches Once the tree is chosen, we have two possible approaches to solve the problem: 1. Cutset-voltage approach (c.f. nodal) 2. Loop-current approach (c.f. mesh) Unknowns are tree voltages Set up KCL equations based on basic cutsets Unknowns are co-tree (link) currents Set up KVL equations based on basic loops
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Prof. C.K. Tse: Graph Theory & Systematic Analysis 31 Cutset-voltage approach Step 1: Start with the digraph. Choose a tree. Define unknowns as the tree voltages. Label all voltages. – + + 2A 1V 3V 2S 2S 1S 2S 1S + V 1 V 2 + 1 2 1 2 3 4 5 Step 2: Write the KCL equations for each basic cutset (except those corresponding to voltage sources) Cutset 1: Cutset 2:
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Prof. C.K. Tse: Graph Theory & Systematic Analysis 32 Loop-current approach Step 1: Start with the digraph. Choose a tree. Define unknowns as the co-tree currents. Label all currents. Step 2: Write the KVL equations for each basic loop (except those corresponding to current sources) Loop 1: Loop 2: + 7V 2Ω 2Ω 1Ω 1Ω 7A 3Ω 1 2 3 4 5
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Prof. C.K. Tse: Graph Theory & Systematic Analysis 33 Choice of method Cutset-voltage method: Equations to be solved = t – (number of voltage sources) = n – 1 – (number of voltage sources) Loop-current method: Equations to be solved = b – t – (number of current sources) = b – n + 1 – (number of current sources) CHOOSE THE SIMPLEST!
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Prof. C.K. Tse: Graph Theory & Systematic Analysis 34 Question!! So far, we have only focused on finding EITHER the tree voltages OR the co-tree currents How about other branch currents and voltages? Can you verify the following: Once we know either the tree voltages or the co-tree currents, we can derive everything else in the circuit.
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Prof. C.K. Tse: Graph Theory & Systematic Analysis 35 Sherlock Holmes’ search Tree: Voltage sources Resistors Co-tree: Resistors Current sources voltage current ? ? ? ? ? Tree: Voltage sources Resistors Co-tree: Resistors Current sources voltage current ? ? ? ? ? ? Cutset-voltage method: Loop-current method: KVL B-loop KVL B-loop Ohm’s law Ohm’s law KCL B-cutset KCL B-cutset Ohm’s law Ohm’s law KCL B-cutset KVL B-loop
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Prof. C.K. Tse: Graph Theory & Systematic Analysis 36 Conclusion n Graph theory n Take advantage of topology n Cutset-voltage approach n Aim to find all tree voltages initially n Loop-current approach n Aim to find all cotree currents initially
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  • Summer '16
  • Martin Chow
  • Graph Theory, Mesh Analysis, branch, Voltage source, Voltage drop, Systematic Analysis

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