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1 Nodal Analysis Discussion D2.3 September 2006 Chapter 2 Section 2-7

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2 Nodal Analysis Interested in finding the NODE VOLTAGES, which are taken as the variables to be determined For simplicity we start with circuits containing only current sources

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3 Nodal Analysis Steps 1.Select one of the n nodes as a reference node (that we define to be zero voltage, or ground). Assign voltages v 1, v 2, … v n-1 to the remaining n-1 nodes. These voltages are referenced with respect to the reference node. 2.Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of the node voltages. 3.Solve the resulting simultaneous equations to obtain the node voltages v 1, v 2, … v n-1.

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4 Example Select a reference node as ground. Assign voltages v 1, v 2, and v 3 to the remaining 3 nodes.

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5 Example Apply KCL to each of the 3 non-reference nodes (sum of currents leaving node is zero). Node 1: Node 2: Node 3:

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6 Example Now express i 1, i 2, …i 5 in terms of v 1, v 2, v 3 (the node voltages). Note that current flows from a higher to a lower potential.

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7 Node 1: Node 2: Node 3:

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8 In MATLAB, if then is a row matrix of the five conductances

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9 Node 1: Node 2: Node 3:

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10 These three equations can be written in matrix form as

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11 is an (n –1) x (n –1) symmetric conductance matrix is a 1 x (n-1) vector of node voltages is a vector of currents representing “known” currents

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12 Writing the Nodal Equations by Inspection The matrix G is symmetric, G kj = G jk and all of the off-diagonal terms are negative or zero. The k i (the i th component of the vector k) = the algebraic sum of the independent currents connected to node i, with currents entering the node taken as positive. The G kj terms are the negative sum of the conductances connected to BOTH node k and node j. The G kk terms are the sum of all conductances connected to node k.

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13 MATLAB Solution of Nodal Equations

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14 v1v1 v2v2 v3v3 Test with numbers

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15 MATLAB Run v1v1 v2v2 v3v3

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16 PSpice Simulation MATLAB:

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17 Let's write a general MATLAB program to solve this problem Inputs: Find all voltages and currents

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18 function nodal1(r,k) % PowerPoint nodal example % Discussion D2.3 % r is a 1 x 5 vector of resistances % k is a 3 x 1 vector of known currents entering the three nodes % nodal1(r,k) % g = 1./ r G = [g(1)+g(2) -g(2) 0; -g(2) g(2)+g(3)+g(4) -g(4); 0 -g(4) g(4)+g(5)] k v = inv(G)*k i(1) = v(1)*g(1); i(2) = (v(1) - v(2))*g(2); i(3) = v(2)*g(3); i(4) = (v(2) - v(3))*g(4); i(5) = v(3)*g(5); i kab = [i(1)+i(2) i(5)-i(4)]

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19 Do same problem as before nodal1(r,k)

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20 MATLAB Run

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21 Nodal Analysis for Circuits Containing Voltage Sources That Can’t be Transformed to Current Sources Case 1. If a voltage source is connected between the reference node and a nonreference node, set the voltage at the nonreference node equal to the voltage of the source. Case 2. If a voltage source is connected between two nonreference nodes, assume temporarily that the current through the voltage source is known and write the equations by inspection.

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22 Example Assume temporarily that i 2 is known and write the equations by inspection.

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23 There appears to be 4 unknowns (v 1, v 2, v 3, and i 2 ) and only 3 equations. However, from the circuit or so we can replace v 1 (we could also replace v 2 ) and write

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24 Writing the above equation with the unknowns (v 2, v 3, i 2 ) on the LHS yields

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25 v1v1 v2v2 v3v3 Test with numbers Noting that

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26 v1v1 v2v2 v3v3 Test with numbers Unknowns:

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27 MATLAB Run V V A v1v1 v2v2 v3v3 v2v2 v3v3 i2i2

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28 PSpice Simulation MATLAB: v2v2 v3v3 i2i2

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29 Let's write a general MATLAB program to solve this problem Inputs: Find all voltages and currents

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30 function nodal2(g,V0,is) % PowerPoint nodal-2 example % Discussion D2.3 % g is a 1 x 4 vector of conductances % V0 = the known dc voltage source % is = the known dc current source % nodal2(g,V0,Is) % G = [g(1) 0 1; g(2)+g(3) -g(3) -1; -g(3) g(3)+g(4) 0] k = [-2+g(1)*V0; 0; is] vvi = inv(G)*k v = zeros(1,3); v(2) = vvi(1); v(3) = vvi(2); v(1) = v(2)-V0; v i(1) = v(1)*g(1); i(2) = vvi(3); i(3) = v(2)*g(2); i(4) = (v(2) - v(3))*g(3); i(5) = v(3)*g(4); i kab = [i(1)+i(2) i(5)-i(4)]

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31 Do same problem as before nodal2(g,V0,is)

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32 MATLAB Run

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