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Nodal Analysis: Grounded voltage sources

+. 3 W. 2 W. 4 W. 1 W. B. 5 W. 6 W. D. C. A. V in. I in. G = ref. Nodal Analysis: Grounded voltage sources. Essential Nodes : connect more than 2 elements A, C, D, G  3 independent equations to solve for all node voltages (V A , V B , V C )

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Nodal Analysis: Grounded voltage sources

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  1. + 3 W 2 W 4 W 1 W B 5 W 6 W D C A Vin Iin G = ref Nodal Analysis: Grounded voltage sources • Essential Nodes: connect more than 2 elements A, C, D, G  3 independent equations to solve for all node voltages (VA, VB, VC) • VA = Vin 2 equations to solve for VC and VD • Theorem: For a circuit with n nodes, applying KCL to (n-1) nodes yields (n-1) linearly independent equations. • Apply KCL at C and D to get 2 linearly independent equations

  2. B + D C A 2 W 4 W 1 W 3 W 5 W 6 W Vin Iin Nodal Analysis: Grounded Vs (Cont.) • At each node label all currents out: @C: @D: Inversion Method: Cramer’s Rule:

  3. Linear System of Equations • Let A be an (nxn) matrix, and v an (nx1) vector: Ax = v has a unique (nx1) vector solution x = A-1v iff A-1exists i.e. det(A)  0. • det(A)  0  Ax = v forms n linearly independent equations • Computing determinant: 2x2 matrix: 3x3 matrix:

  4. Methods to Solve Linear System of Eq. Solve for a (nx1) vector x in Ax = v • Inverse method:x = A-1v Computing inverse matrix: • 2x2: • 3x3 of higher: ijth element of AA = (-1)i+j Dij, Dij = det of matrix formed by deleting ith row and jth column of A. • Gauss method of elimination and its variations. • Cramer’s Rule: where D = det(A), Dk = det(Ak) and Ak matrix formed by replacing the kth column of A by v

  5. 0.3 Vx Rin + Vx - + Vs - 2W Iy 4 Iy Is 4 W + 10 W 4 W Drill • Determine Rin: Is= 4A. Dependent sources  Need Vs Nodal Analysis • Essential Nodes: • Unknown node voltages:

  6. + + Nodal Analysis: floating voltage sources a • Essential nodes: a, b, c, GND • Unknown node voltages: Vb and Vc  2 equations • Equation 1: KCL @ super node • Equation 2: terminal node voltages of the floating voltage source 10W 90 W 50ix 50V ix c b Super node 10 W 90 W

  7. Steps for Nodal Analysis 1. Choose a reference node and identify essential nodes. 2. Discard essential nodes that are connected to the reference node through a path made of one or more voltage sources. 3. Remaining nodes = unknown node voltages (variables to solve for). 4. Circle floating voltage sources to form super nodes 5. Write KCL at each super node and each unknown node voltage not included in a super node. 6. Write a terminal equation for each super node in term of unknown node voltages. 7. If controlled sources are present, then write an equation that expresses the control current/voltage in terms of unknown node voltages.

  8. + B 3 W 2 W 4 W 1 W I2 D C A I1 Iv Iin 5 W 6 W Vin Limitations of Nodal Analysis • List the steps needed to find Iv by nodal analysis: • find the node voltages VD and VC • compute • Iv = I1 + I2 • Need circuit analysis method to find current directly.

  9. + 1 W B A 4 W I2 2 W Iin Ia C D 2 W I1 + Vin 6V Ib I3 1 W 1 W G Mesh Analysis • Loop: closed current path. • Mesh: a loop that does not contain any loop within it. • Mesh currents  all V and I for each circuit element • Mesh currents: • Real? • Measurable? • Branch currents: • Iin = I1 • Ia = I3 - I2 • Ib = I3 - I1 • Theorem: For a circuit with n nodes, and b branches, the maximum number of linearly independent equations based on KVL is (b-n+1).

  10. + 1 W B A 4 W I2 2 W C D 2 W I1 + Vin 6V I3 1 W 1 W Applying Mesh Analysis • Identify meshes  3 mesh currents, I1, I2 and I3. • Need 3 independent equations  write KVL around each mesh.

  11. 10A + I2 3 W 6 W 12 W I1 30 W 240V I3 60V Iin G + Current Source in Outer Meshes • Find Iin? • 3 meshes: L1, L2 and L3 • I2 = 10A 2 mesh currents to solve for

  12. + Current Source Between Meshes #17 p.115: • 3 meshes 3 mesh currents 3 equations • To apply KVL @ L1 and L2 we need the voltage across the 6A current source write KCL @ Super mesh I2 2 W 1W I1 3 W 6A 7V 1 W I3 2W

  13. Steps for Mesh Analysis 1. Identify all meshes. 2. Discard border meshes that contain an outer current source. 3. Remaining meshes = unknown mesh currents (variables to solve for). 4. Form super meshes to avoid current sources shared by 2 meshes. 5. Write KVL around each super mesh and each unknown unknown mesh not included in a super mesh. 6. Write the current of each shared current source in term of unknown mesh currents. 7. If controlled sources are present, then write an equation that expresses the control current/voltage in terms of unknown mesh currents.

  14. Drill • Use the steps in the previous slide to solve #11 p. 113

  15. Modified Nodal Analysis (MNA) • Nodal analysis solves only for node voltages • MNA solves for node voltages & auxiliary currents • Auxiliary currents are currents through voltage sources and shorts, control currents and output currents. • Steps for MNA: 1. Choose a reference node and identify essential nodes. 2. Discard essential nodes connected to grounded voltage source(s) as long as they have no auxiliary current(s). 4. Circle floating voltage sources that have no auxiliary current(s) to form super nodes 5. Write KCL at each super node and each unknown node voltage that was not discarded in (2) 6. Write a terminal equation for each super node in term of unknown node voltages. 7. Write a terminal equation for each branch containing an auxiliary current.

  16. Nodal vs. Mesh Analysis • For computers: easier to identify nodes than meshes. • For non-planar circuits ( circuits with crossings): difficulty defining meshes (Example 4.11 p. 108). • For a circuit with n nodes and b branches: • Nodal analysis: • KCL at (n-1) nodes • (n-1) linearly independent equations • solve for (n-1) variables. • Mesh analysis: • KVL at (b-n+1) nodes • (b-n+1) linearly independent equations • solve for (b-n+1) variables.

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