ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 1 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 43 Chp 3.1a Nodal Analysis

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 2 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Chapter-3 Learning Goals  Nodal Analysis Develop Systematic Techniques To Determine All The Voltages In A Circuit –Can Then Find Branch Currents by Ohm’s Law  Loop Analysis Develop Systematic Techniques To Determine All The Currents In A Circuit –Can Then Find Node Voltages by Ohm’s Law

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 3 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Nodal Analysis  A Systematic Technique To Determine Every Voltage And Current In A Circuit  The variables used to describe the circuit will be “Node Voltages” The voltages of each node Will Be Determined With Respect To a Pre-selected REFERENCE Node –The Reference Node is Often Referred to as  Ground (GND)  Or  COMMON

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 4 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Consider Resistor Ladder  Goal: Determine All Currents & Potentials In this “Ladder” Network  Plan Use Series/Parallel Transformation to Find I 1 Back-Substitute Using KVL, KCL, Ohm to Find Rest

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 5 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Series-Parallel Transformations  Xform1 Combine 3 Resistors at End of Network  Xform2 Combine 3 Resistors at End of Network  Note By Ohm’s Law

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 6 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Xform cont.  Xform3 To Single-Loop Ckt  Now Back Substitute Recall By KCL

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 7 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Xform cont.  Recall Xform2  In Summary I 1 = 1 mA I 2 = I 3 = 0.5 mA I 4 = mA I 5 =0.125 mA V a = 3 V V b = 1.5 V V c = V  Finally by KCL

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 8 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Node Analysis Perspective KVL REFERENCE   In General: V x5 = V x − V 5 = V x − 0 = V x Then the KVL Eqns  Take Node-5 As the Ref, →V 5 = 0, Always

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 9 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Node Analysis cont  If We Know V a, V b, and V c, Then Can Calc V 1, V 2, V 3 by KVL, Then –Use Ohm’s Law to Find I 1 →I 5  i.e., If we Know All Node Potentials, Then Can Calc All Branch Currents Theorem: IF ALL NODE VOLTAGES WITH RESPECT TO A COMMON REFERENCE NODE ARE KNOWN, THEN ONE CAN DETERMINE ANY OTHER ELECTRICAL VARIABLE FOR THE CIRCUIT

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 10 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Recall Passive Sign Convention  If V’ Drops R←L i’ by Passive Sign Convention  If V Drops L→R i by Passive Sign Convention

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 11 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis ALWAYS Define Reference Node  The Statement V 1 = 4V is Meaningless UNTIL The Designation of a REFERENCE NODE  By Convention The Ground (GND) Symbol Indicates the Reference Point ALL Node Voltages are Measured Relative to GND

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 12 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Strategy for Node Analysis 1.Identify All Nodes And Select A Ref. Node REFERENCE 2.Identify Known Node Voltages 3.at Each Node With Unknown Voltage Write A KCL Equation e.g.,  (Sum Of Current Leaving) =0 4.Replace Currents In Terms Of Node V’s  Yields Algebraic Eqns In The Node Voltages Final Desired Eqn Set

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 13 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Node Equation Mechanics  When Writing Node Equations At Each Node We Can Choose Arbitrary Directions for Currents Then select any form of KCL  When The Currents Are Replaced In Terms Of The Node Voltages The Node Eqns That Result Are The Same Or Equiv.

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 14 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Node Eqn Mechanics cont.  When Writing The Node Equations Use Ohm’s Law to Write The Equation Directly In Terms Of The Node Voltages BY Default Use KCL In The Form Sum-of- currents-leaving = 0 –But The Reference Direction For The Currents Does NOT Affect The Node Equation

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 15 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ckts w/ Independent Sources  At Node-1 Using Resistances  Using Conductances Eliminates Tedious Division Operations  Replacing R’s w/ G’s At Node-1 At Node-2

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 16 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Node Analysis of Indep Src Ckts  ReOrder Terms in Eqns for i A & i B  The Manipulation Of Systems Of Algebraic Equations Can Be Efficiently Done Using Matrix Analysis c.f., MTH-6 or ENGR-25 (MATLAB)  The Model For The Circuit Is A System Of Algebraic Equations

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 17 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example  Write the KCL Node-1 We Visualize The Currents Leaving And Write the KCL Eqn  Similarly at Node-2 Could Use  (i Entering Node) Just as well

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 18 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis KCL Eqn Example  Write KCL At Each Node In Terms Of Node Voltages 3 Nodes Implies 2 KCL Equations Mark the nodes (to insure that None is missing) Select as Reference  Then

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 19 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Linear Algebra Analysis  Recall R=1/G, Then Insert Numerical Values, and Change to Time Independent Notation (All CAPS)  The Math Model  The Node Eqns in Conductance Form

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 20 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Linear Algebra Analysis cont.  The Numerical Model  Multiply the 1 st Eqn by 4kΩ to Find V 1 in Terms of V 2  Back Sub into 2 nd eqn  Then V 2  And V 1  Alternatively, Multiply Both Sides of Math Model by LCD in kΩ R.H.S. of Eqn Now in Volts V 1, V 2 CoEffs are No.s

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 21 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Linear Algebra Analysis cont.  The “Clean” Eqns  Proceed with Gaussian Elimination Add Eqns to Eliminate V 2 Back Substitute to Find V 2

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 22 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Use Matrix Algebra  Recall The Math Model  From MTH-6 the Form of Matrix Multiplication In this Case  The Matrix Eqn Soln In this Case  Calculating the Matrix Inverse, G -1, is NOT Trivial Use Matrix Manipulation –Adjoint Matrix –Determinant Calculation Or use MATLAB

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 23 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Matrix Algebra cont.  Given A, Find A -1  Adjoint_Matrix_0308.pdf  For The Adjoint Replace Each Element By Its Cofactor Let G = A, Then in this Case  The Determinant  Do The Matrix Algebra

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 24 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis GV = I  By MATLAB  Construct the Coefficient Matrix G >> G = [1/4e3 -1/6e3; -1/6e3 1/3e3] G = 1.0e-003 *  Construct the Constraint Vector, I >> I = [1e-3; -4e-3] I =  Matrix Inversion by “Left” Division for V >> V = G\I V =

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 25 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Nodal Analysis: Indep I src Example  ReArrange Terms 123

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 26 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Nodal Analysis Example cont.  Cast in Matrix Form  Could Write the Equation by Inspection

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 27 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Write Eqns by Inspection Conductances between 1 and 2 Conductances between 1 and 3 Conductances connected to node 1  For Circuits With ONLY INDEPENDENT I-Sources, The Matrix Is ALWAYS Symmetric The Diagonal Elements Are Positive The Off-diagonal Elements Are Negative Conductances between 2 and 3

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 28 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis I src & R Eqns By Inspection  Then Finally The Matrix Multiplication  Note the Symmetrical Form of the Matrix Diagonal Symmetry Occurs for Circuits That Contain ONLY –INDEPENDENT CURRENT Sources –RESISTORS

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 29 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Isrc & R Circuit  Write the Node Equations  By KCL I out = POS  By “Inspection” Method for I src & R Ckts

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 30 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example  Find All BRANCH Currents  By KCL I out = POS  By Inspection  By I-Divider  Most Times have More Than One Solution Path

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 31 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Dependent Current Sources  Circuits With DEPENDENT Sources Can NOT Be Modeled By Inspection The symmetry is LOST  Math Model Construction Write The Node Equations Using Dependent Sources As Regular Sources Each Dependent Source Adds One Equation Expressing The Controlling Variable In Terms Of Node Voltages  In This Case The Node Eqns  Controlling Variable, i o, Model  Sub For i o & Rearrange

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 32 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Dep I src Numerical Example  The Node Eqns i o = 2(v 2 /3kΩ) (as  = 2)  Or  Multiply by LCDs  Adding Eqns (Simplest Gaussian Elimination)

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 33 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ckt w/ V-controlled I src  Write Node Equations  Treat Dependent Source As a Regular Source Node Eqns  Express Controlling Variable In Terms Of Node Voltages

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 34 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example Ckt w/ V-controlled I src  4 Eqns in 4 Unknowns Solve Using Most Convenient Method –Choose SUB & GAUSSIAN ELIM  Sub for v x in v x I src  Continue w/ Gaussian Elim OR Use Matrix Algebra

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 35 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Solve Using MATLAB  Define Components (m-file Node_Anal_0602.m) R1 = 1000; R2 = 2000; R3 = 2000; R4 = 4000; %resistances in Ohms iA = 0.002; iB = 0.004; %sources in Amps Alpha = 2; %gain of dependent source in Siemens

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 36 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Solve Using MATLAB cont  Define Coefficient Matrix G=[(1/R1+1/R2), -1/R1, 0; % first Matrix row -1/R1,(1/R1+alpha+1/R2),-(alpha+1/R2); % 2nd row 0, -1/R2,(1/R2+1/R4)] %third row. G =

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 37 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Solve Using MATLAB cont  Define Constraint Vector I=[iA;-iA;iB];  Solve by Left/Back Division; V in volts V=G\I % end with carriage return and get the ReadBack V =

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 38 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Example  Find Node Potentials The Node Eqns  Controlling Variable In Terms of Voltages  Sub for I o  LCD-Mult & Rearranging  Multiply Top Eqn by 2, Then Add Eqns

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 39 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Voltage Output Example  Find V o  Node Eqns –Note Replacement Of Dep. Src In Terms Of Node Voltage  Multiply Eqns by LCDs 6 kΩ 12 kΩ

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 40 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 43 Appendix Adjoint Matrix Construction

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 41 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Construct Adjoint Matrix  Consider a 2X2 Matrix  Construct Adjoint Matrix in 3 Steps 3.Take the Determinant Minors –Minor for 2 –Minor for -7 –Minor for 5 –Minor for 3 1.Take Transpose by Switching Rows & Columns 2.The Sign Convention for Adjoint

ENGR-44_Lec-03-1a_Nodal_Analysis.ppt 42 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Construct Adjoint Matrix cont  Build Adjoint Using Steps 2&3  Thus The Adjoint For Matrix G in Text