ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 1 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 43 MaxPower SuperPosition

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 2 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis OutLine: MaxPwr & SuperPose  Work On WhtBd Student Suggest HomeWork Problem  Thevénin & Norton Review Example Problem (WhtBd)  Maximum Power Transfer Theorem Derivation MaxPwr Application Examples  Thevénin & Norton Summary

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 3 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis OutLine: MaxPwr & SuperPose  Linearity & Homogeneity Guess Solution, Work BackWards, Scale Guess Comparative Case Study  SuperPosition → Activate & DeActivate Example Problem (WhtBd)

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 4 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Thevénin’s Equivalence Theorem  Thevenin Equivalent Circuit for PART A  v TH = Thévenin Equivalent VOLTAGE Source  R TH = Thévenin Equivalent SERIES RESISTANCE

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 5 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Norton’s Equivalence Theorem  Norton Equivalent Circuit for PART A  i N = Norton Equivalent CURRENT Source  R N = Norton Equivalent PARALLEL RESISTANCE

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 6 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example: V OC, I SC, R TH = R N  Use Thevénin and Norton for find the OutPut Voltage in the Circuit Below  Recall: V TH = V OC & I N = I SC

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 7 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 8 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 9 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Now Isc

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 10 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 11 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 12 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 13 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Maximum Power Transfer  Consider The Amp-Speaker Matching Issue From PreAmp (voltage ) To speakers

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 14 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Maximum Power Xfer Cont  The Simplest Model for a Speaker is to Consider it as a RESISTOR only  Since the “Load” Does the “Work” We Would like to Transfer the Maximum Amount of Power from the “Driving Ckt” to the Load Anything Less Results in Lost Energy in the Driving Ckt in the form of Heat BASIC MODEL FOR THE ANALYSIS OF POWER TRANSFER

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 15 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Maximum Power Transfer  Consider Thevenin Equivalent Ckt with Load R L  Find Load Pwr by V-Divider  For every choice of R L we have a different power. How to find the MAXIMUM Power value?  Consider P L as a FUNCTION of R L and find the maximum of such a function  have at left! i.e., Take 1 st Derivative and Set to Zero

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 16 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Power Xfer cont  Find Max Power Condition Using Differential Calculus  Set The Derivative To Zero To Find MAX or MIN Points For this Case Set To Zero The NUMERATOR  Solving for “Best” (P max ) Load  This is The Maximum Power Transfer Theorem The load that maximizes the power transfer for a circuit is equal to the Thevenin equivalent resistance of the circuit

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 17 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Power Quantified  By Calculus we Know R L for P L,max  Recall the Power Transfer Eqn  Sub R TH for R L  So Finally

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 18 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Pwr Xfer Example  Determine R L for Maximum Power Transfer  Need to Find R TH Notice This Ckt Contains Only INDEPENDENT Sources  Thus R TH By Source Deactivation  This is Then the R L For Max Power Transfer  To Find the AMOUNT of Power Transferred Need the Thevenin Voltage  Then use R TH = 6kΩ along with V TH ab

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 19 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Pwr Xfer Example cont  To Find V TH Use Meshes  The Eqns for Loops 1 & 2  Solving for I 2  Now Apply KVL for V OC  Recall  At Max: P L = P MX, R L = R TH

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 20 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Pwr Xfer  Determine R L and Max Power Transferred  Find Thevenin Equiv. At This Terminal-Set  Recall for Max Pwr Xfer a b  This is a MIXED Source Circuit Analysis Proceeds More Quickly if We start at c-d and Adjust for the 4kΩ at the end c d  Use Loop Analysis  Eqns for Loops 1 & 2

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 21 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Max Pwr Xfer cont  The Controlling Variable  Now Short Ckt Current The Added Wire Shorts the 2k Resistor  Remember now the partition points  Then R TH c d  The R TH for ckt at a-b = 2kΩ+4kΩ; So a b

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 22 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Thevenin & Norton Summary  Independent Sources Only R TH = R N by Source Deactivation V TH –= V OC or –= R N ·I SC I N –= I SC or –= V OC /R TH  Mixed INdep and Dep Srcs Must Keep Indep & dep Srcs Together in Driving Ckt V TH = V OC I N = I SC R TH = R N = V OC / I SC  DEPENDENT Sources Only Must Apply V or I PROBE –Pick One, say I P = 1.00 mA, then Calculate the other, say V P V TH = I N = 0 R TH = R N = V P / I P

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 23 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis WhiteBoard Work LLet’s Work this nice Max Power Problem FFind P max for Load R L

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 24 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Previous Equivalent Circuits   [Independent Srcs] V src ’s in Series I src ’s in Parallel  Series & Parallel Resistors  The Complementary Configs are Inconsistent with Source Definitions

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 25 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Linearity  Models Used So Far Are All LINEAR Mathematically This Implies That They satisfy the principle of SUPERPOSITION  The Model T(u) is Linear IF AND ONLY IF For All Possible –Input Pairs: u 1 & u 2 –Scalars α 1 & α 2  AN Alternative, And Equivalent, Linearity & Superposition Definition The Model T(u) is Linear IF AND ONLY IF It Exhibits –ADDITIVITY –HOMOGENEITY

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 26 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Linearity cont.  Linearity Characteristics Additivity  NOTE Technically, Linearity Can Never Be Verified Empirically on a System But It Could Be Disproved by a SINGLE Counter Example. It Can Be Verified Mathematically For The Models Used Homogeneity –a.k.a. Scaling

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 27 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Linearity cont.  Using Node Analysis For Resistive Circuits Yields Models Of The Form  The Model Can Be Made More Detailed Where –A and B are Matrices –s Is A Vector Of All Independent Sources  For Ckt Analysis Use The Linearity Assumption To Develop Special Analysis Methods Where –v Is A Vector Containing All The Node Voltages – f Is a Vector Containing Only independent Sources

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 28 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Past Techniques  Case Study  Find V o  Redraw the Ckt to Reveal Special Cases After Untangling  Solution Techniques Available?      

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 29 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Case Study cont.  Loop Analysis for V o  Node Analysis Out → Positive

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 30 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Case Study cont.  Series-Parallel Resistor- Combinations  In other Words  By VOLTAGE Divider

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 31 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Use Homogeneity Analysis  Find V o by Scaling  If V o is Given Then V 1 Can Be Found By The Inverse Voltage Divider  Now Use V S As a 2 nd Inverse Divider  Then Solve for V o  Assume That The Answer Is KNOWN How to Find The Input In A Very Easy Way ?

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 32 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Homogeneity Analysis cont  The Procedure Can Be Made Entirely Algorithmic 1.Give to Vo Any Arbitrary Value (e.g., V’o = 1V ) 2.Compute The Resulting Source Value and Call It V’s 3.Use linearity 4.The given value of the source (Vs) corresponds to 5.Then The Desired Output

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 33 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Homogeneity Comment  This is a Nice Tool For Special Problems  Normally Useful When There Is Only One Source Best Judgment Indicates That Solving The Problem BACKWARDS Is Actually Easier Than the Forward Solution-Path

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 34 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration  Homogeneity  Solve Using Homogeneity (Scaling)  Then By Ohm’s Law  Assume V’ out = V 2 = 1volt

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 35 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration  Homogeneity cont  Solve Using Homogeneity  Using Homogeneity Scale from Initial Assumption:  Again by Ohm’s Law  Then  Scaling Factor

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 36 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Source Superposition  This Technique Is A Direct Application Of Linearity  Normally Useful When The Circuit Has Only A Few Sources

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 37 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration  Src Superposition  Consider a Circuit With Two Independent Sources: V S, I S  Calculated By Setting The CURRENT Source To ZERO (OPEN ckt) And Solving The Circuit  Calculated By Setting The VOLTAGE Source To ZERO (SHORT ckt) And Solving The Circuit  Now by Linearity

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 38 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration cont = +  Circuit With Current Source Set To Zero OPEN Ckt  Circuit with Voltage Source set to Zero SHORT Ckt  By Linearity

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 39 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration cont. = +  The Above Eqns Illustrate SUPERPOSITION  This approach will be useful if solving the two, 1-Src circuits is simpler, or more convenient, than solving a circuit with two sources  We can have any combination of sources. And we can partition any way we find convenient

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 40 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example  Solve for i 1 Loop equations Contribution of v1 Contribution of v2  Once we know the “partial circuits” we need to be able to solve them in an efficient manner = +  Alternative for i 1 (t) By SuperPosition: –Find i 1 ’’ by I-Divider

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 41 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Example  Find V o By SuperPosition  Set to Zero The V-Src i.e., SHORT it Current division Ohm’s law Contribution by I src →

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 42 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Example cont.  Find V o By SuperPosition  Set to Zero The I-Src i.e., OPEN it  Yields Voltage Divider (UN- tangle)  Finally, Add by SuperPosition  By V-Divider Contribution by V src

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 43 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis WhiteBoard Work  Let’s Work this Nice SuperPosition Problem FFind I O

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 44 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example  SuperPosition  Find V o Using Source SuperPosition  Set to Zero The I-Src i.e., OPEN it  Set to Zero The V-Src i.e., SHORT it

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 45 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example cont  Define V 1 on the V-Src ckt  If V 1 is known then V’o is obtained using the 6&2 Voltage-Divider V 1 can be obtained by series parallel reduction and divider

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 46 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Example cont.2  Determine Current I 2 By Current Divider V”o Using Ohm’s Law  When in Doubt REDRAW  The Current Division  Finally The SuperPosition Addition

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 47 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Sample Problem  Determine I o by Source SuperPosition  First Consider Only the Voltage Source Yields  Then  Second Consider Only the 3 mA I-Source Yields Current Divider

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 48 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Sample Prob cont  Determine I o by Source SuperPosition  By I O2 Current Divider  Third Consider 4mA Src  The Current will Return on the Path of LEAST Resistance; Thus  So by Source Superposition

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 49 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration  Use Source Superposition to Determine I o  Open the Current Source  By Equivalent Resistance  Next Short the V-Source

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 50 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration cont  Looks Odd & Confusing → REDRAW  Finally By Linearity  Now Use I-Divider

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 51 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis All Done for Today SuperPosition Of Plane-Polarized Light  Red & Green Light-Waves are Polarized in Perpendicular Planes Cyan = Red + Green

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 52 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 53 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 54 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 55 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 56 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 57 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 58 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 59 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 60 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 61 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 62 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 63 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 64 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 65 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 66 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 67 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 43 Appendix: Wheatsone Bridge P2.103

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 68 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis The WheatStone Bridge  When the Wheatstone Bridge is Balanced:

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 69 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 70 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 71 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis

ENGR-43_Lec-02c_Sp12_MaxPwr_SuperPosition.pptx 72 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis