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Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

Engineering 43. Chp 3.1b Nodal Analysis. Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Need Only ONE KCL Eqn. Ckts with Voltage Sources. The Remaining Eqns From the Indep Srcs. 3 Nodes Plus the Reference. In Principle Need 3 Equations...

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Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Engineering 43 Chp 3.1bNodal Analysis Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. Need Only ONE KCL Eqn Ckts with Voltage Sources • The Remaining Eqns From the Indep Srcs • 3 Nodes Plus the Reference. In Principle Need 3 Equations... • But two nodes are connected to GND through voltage sources. Hence those node voltages are KNOWN • Solving The Eqns

  3. Example • Find Vo • To Start • Identify & Label All Nodes • Write Node Equations • Examine Ckt to Determine Best Solution Strategy R1 = 1k; R2 = 2k, R3 = 1k, R4 = 2k Is1 =2mA, Is2 = 4mA, Is3 = 4mA, Vs1 = 12 V • Notice • Need Only V1 and V2to Find Vo • Now KCL at Node 1 • Known Node Potential

  4. Example cont. • At Node 2 • At Node 4 R1 = 1k; R2 = 2k, R3 = 1k, R4 = 2k Is1 =2mA, Is2 = 4mA, Is3 = 4mA, Vs = 12 V • To Solve the System of Equations Use LCD-multiplication and Gaussian Elimination

  5. The LCDs *2k *2k *2k Example cont. (1) (2) (3) • Now Add Eqns (2) & (3) To Eliminate V4 (4) • Now Add Eqns (4) & (1) To Eliminate V2 • BackSub into (4) To Find V2 • Find Vo by Difference Eqn

  6. Consider ThisExample Conventional Node Analysis Requires All Currents At A Node SuperNode Technique SUPERNODE • But Have Ckt V-Src Reln • More Efficient solution: • Enclose The Source, And All Elements In Parallel, Inside A Surface. • Call That a SuperNode • 2 eqns, 3 unknowns...Not Good • Recall: The Current thru the Vsrc is NOT related to the Potential Across it

  7. Apply KCL to the Surface Supernode cont. SUPERNODE • The Source Current Is interior To The Surface And Is NOT Required • Still Need 1 More Equation – Look INSIDE the Surface to Relate V1 & V2 • Now Have 2 Equations in 2 Unknowns • Then The Ckt Solution Using LCD Technique • See Next Slide

  8. The Equations Now Apply Gaussian Elim • Use The V-Source Rln Eqn to Find V2 SUPERNODE • Mult Eqn-1 by LCD (12 kΩ) • Add Eqns to Elim V2

  9. Find the node voltages And the power supplied By the voltage source To compute the power supplied by the voltage source We must know the current through it: @ node-1 BASED ON PASSIVE SIGN CONVENTION THE POWER IS ABSORBED BY THE SOURCE!!

  10. Write the Node Equations KCL At v1 Illustration using Conductances  • At The SuperNode Have V-Constraint • v2−v3= vA • KCL Leaving Supernode   • Now Have 3 Eqnsin 3 Unknowns • Solve Using Normal Techniques

  11. Find Io Known Node Voltages SUPERNODE Example • The SuperNode V-Constraint • Now KCL at SuperNode • Or

  12. Student Exercise • Lets Turn on the Lights for 5-7 min • Students are invited to Analyze the following Ckt • Hint: Use SuperNode • Determine the OutPut Current, IO

  13. Find Io Using Nodal Analysis Known Voltages for Sources Connected to GND SUPERNODE Numerical Example • Now Notice That V2 is NOT Needed to Find Io • 2 Eqns in 2 Unknowns • The Constraint Eqn • Now KCL at SuperNode • By Ohm’s Law

  14. Write the Node Eqns Set UP Identify all nodes Select a reference Label All nodes supernode Complex SuperNode • Nodes Connected To Reference Through A Voltage Source • Eqn Bookkeeping: • KCL@ V3 • KCL@ SuperNode, • 2 Constraint Equations • One Known Node • Voltage Sources In Between Nodes And Possible Supernodes • Choose to Connect V2 & V4

  15. Now KCL at Node-3 supernode Complex SuperNode cont. Vs2 Vs3 • Now KCL at Supernode • Take Care Not to Omit Any Currents Vs1 • Constraints Due to Voltage Sources • 5 Equations 5 Unknowns → Have to Sweat Details

  16. Dependent Sources • Circuits With Dependent Sources Present No Significant Additional Complexity • The Dependent Sources Are Treated As Regular Sources • As With Dependent CURRENT Sources Must Add One Equation For Each Controlling Variable

  17. Find Io by Nodal Analysis Notice V-Source Connected to the Reference Node Numerical Example – Dep Isrc • Sub Ix into KCL Eqn • KCL At Node-2 • Mult By 6 kΩ LCD • Controlling Variable In Terms of Node Potential • Then Io

  18. Find Io by Nodal Analysis Notice V-Source Connected to the Reference Node Dep V-Source Example • SuperNode Constraint • KCL at SuperNode • Controlling Variable in Terms of Node Voltage • Mult By 12 kΩ LCD

  19. Simplify the LCD Eqn Dep V-Source Example cont • By Ohm’s Law

  20. Find Io Supernode Constraint Current Controlled V-Source • Controlling Variable in Terms of Node Voltage • Multiply by LCD of 2 kΩ • Recall • Then • KCL at SuperNode • So Finally

  21. IX WhiteBoard Work • Let’s Work This Problem • Find the OutPut Voltage, VO

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